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Certain Classes of Codes fromCombinatorial Designs
Yin Jianxing
Department of Mathematics,
Suzhou University
September 14, 2006
J. Yin, Sept. 14, 20061
Japan Seminar Codes From Combinatorial Designs
1.1 Constant Composition Codes
• An (n,M, d; q)-code is a code C ⊆ Qn with length n, size
M(= |C|) and Hamming distance d(C) = d.
• Here, Q is an alphabet of q elements called symbols or letters.
Most of the time Q is taken to be the finite field GF(q) of order q
or the ring of integers modulo q.
• An (n,M, d; q)-code is refereed to as a constant weight code
(CWC), or an (n,M, d, w; q)-CWC, if its codewords have the
same weight w.
J. Yin, Sept. 14, 20062
Japan Seminar Codes From Combinatorial Designs
• Let Q = {at : 0 ≤ t ≤ q − 1}. An (n,M, d; q)-code over Q is
refereed to as a constant composition code (CCC), or an
(n,M, d, [w0, w1, · · · , wq−1]; q)-CCC, if for any i (0 ≤ i ≤ q − 1),
the symbol ai appears exactly wi times in every codeword.
• The constant composition [w0, w1, · · · , wq−1] is called the type
of the CCC, which is essentially an unordered multiset. We will
write it in an exponential notation in the sequel. In case the wi
are themselves exponents, we revert to the composition list to
avoid confusion.
J. Yin, Sept. 14, 20063
Japan Seminar Codes From Combinatorial Designs
• CCCs arise in frequency hopping, when a schedule is needed to
determine frequencies on which to transmit. When each frequency
is to be used a specified number of times within a frame, each
frequency hopping sequence is a codeword of constant compo-
sition. Indeed whenever a different cost is associated with each
symbol in the underlying alphabet, uniform cost of codewords
leads to constant composition specified number of time, see Chu,
Colbourn and Dukes (DAM 2006).
• CCCs are also useful in the powerline communication.
J. Yin, Sept. 14, 20064
Japan Seminar Codes From Combinatorial Designs
• Constant composition codes are also a subclass of constant
weight codes by definition.
• The class of binary constant composition codes coincides with the
class of binary constant weight codes
• An (n,M, d, [w0, w1, . . . , wq−1]; q)-CCC is called a permutation
code if n = q and wi = 1 for all i. Hence, permutation codes are
a special class of constant composition codes.
J. Yin, Sept. 14, 20065
Japan Seminar Codes From Combinatorial Designs
1.2 Bounds on CCCs
• One of the most fundamental problems in combinatorial
coding theory is to determine the maximum size of a
block code when its other parameters have been fixed.
• The notation A(n, d, [w0, w1, ..., wq−1]; q) stands for the maximum
size of an (n,M, d, [w0, w1, ..., wq−1]; q) constant composition code.
A CCC with A(n, d, [w0, w1, ..., wq−1]; q) codewords is said to be
optimal.
J. Yin, Sept. 14, 20066
Japan Seminar Codes From Combinatorial Designs
• To measure the optimality of CCCs, there are two classic upper
bounds that we can employ: the Johnson and the Plotkin bounds.
The latter holds for any code.
• Plotkin Bound: If an (n,M, d; q)-code with d > n(q − 1)/q
exists, then
M ≤ qd
qd− n(q − 1).
J. Yin, Sept. 14, 20067
Japan Seminar Codes From Combinatorial Designs
• The Johnson bounding technique (Johnson (1962)) was
applied to ternary CCCs by Svanstrom et al. (IEEE IT 2002):
• Johnson Bound: For any integer r satisfying 0 ≤ r ≤ q − 1,
A(n, d, [w0, w1, · · · , wq−1]; q) ≤ n
wr
A(n− 1, d, [w0, w1, · · · , wq−1]; q)
where
wi =
{wi − 1, if i = r
wi, if i 6= r.
J. Yin, Sept. 14, 20068
Japan Seminar Codes From Combinatorial Designs
• Luo, Fu, Han Vink and Chen (IEEE IT 2003) established the
following bound for CCCs.
LFVC Bound: If nd− n2 + (w20 + w2
1 + · · ·+ w2q−1) > 0, then
A(n, d, [w0, w1, ..., wq−1]; q) ≤ nd
nd− n2 + (w20 + w2
1 + · · ·+ w2q−1)
.
J. Yin, Sept. 14, 20069
Japan Seminar Codes From Combinatorial Designs
• (Ding and Yin (IEEE IT 2005-2)) Let
N =(w2
0 + w21 + · · ·+ w2
q−1
)− λn;
λ = gcd {wi| i = 0, 1, · · · , q − 1};N = Nλ.
If N > 0, n(n− λ) ≡ 0 (mod N) and n− λ 6≡ 0 (mod N), then
A(n, n− λ, [w0, w1, · · · , wq−1]) ≤ n(n− λ)
N− 1.
J. Yin, Sept. 14, 200610
Japan Seminar Codes From Combinatorial Designs
• Combining LFVC Bound with the Johnson Bound produces
the following.
Lemma 1.1 For any (n,M, n− λ, [λ, λ, ..., λ]; q)-CCC,
M ≤ q(n− 1),
where n = λq.
J. Yin, Sept. 14, 200611
Japan Seminar Codes From Combinatorial Designs
1.3 The Comb. Characterization
• Suppose that there is a set X of v points and that from these
a collection A of subsets, or blocks, is drawn. The ordered pair
(X,A) is called a design of order v.
• In design theory there are normally a number of additional rules
imposed when the blocks are selected.
• A design is called an (n, λ)-packing if there exists a (minimum)
constant λ such that every pair of distinct points occurs in at
most λ blocks, and every point occurs in precisely n blocks.
J. Yin, Sept. 14, 200612
Japan Seminar Codes From Combinatorial Designs
• An α-parallel class of a design is a set of blocks such that each
point occurs in precisely its α blocks. When α = 1, it is simply
termed a parallel class.
• A resolution of the design is a partition of its blocks into
α-parallel classes for certain values of α.
• Two resolutions of a design are said to be orthogonal if any class
in one resolution intersects every class from the other resolution in
at most one block. Here, repeated blocks (if they exist) are
regarded as distinct blocks.
J. Yin, Sept. 14, 200613
Japan Seminar Codes From Combinatorial Designs
• A generalized doubly resolvable packing (GDRP), or a
GDRP(n, λ; v) of type {λ0, λ1, · · · , λm−1} is an (n, λ)-packing of
order v whose blocks can be arranged into an m× n array R with
the following properties.
i. Each cell of R is either empty or contains one block.
ii. For 0 ≤ i ≤ m− 1, the blocks in row i of R form a λi-parallel
class.
iii. The blocks in every column of R form a parallel class.
J. Yin, Sept. 14, 200614
Japan Seminar Codes From Combinatorial Designs
• The type {λ0, λ1, · · · , λm−1} is described by an exponential
notation as with CCCs.
• The definition implies n =∑m−1
i=0 λi. Hence, whenever the
exponential notation ga11 ga2
2 · · · gass is used, we have
n = a1g1 + a2g2 + · · ·+ asgs and m = a1 + a2 + · · ·+ as.
J. Yin, Sept. 14, 200615
Japan Seminar Codes From Combinatorial Designs
• CCCs can be characterized by GDRPs:
Theorem 1.2 (Ding and Yin (IEEE IT 2005-2))
The existence of a GDRP(n, λ; v) of type {λ0, λ1, · · · , λq−1} is
equivalent to that of an (n,M, d, [w0, w1, · · · , wq−1]; q)-CCC,
where
M = v, d = n− λ and λj = wj, 0 ≤ j ≤ q − 1.
J. Yin, Sept. 14, 200616
Japan Seminar Codes From Combinatorial Designs
Example:
A (10, 10, 8, [4123]; 4)−CCC
0 3 2 3 1 0 2 1 0 0 0
1 0 3 2 3 1 0 2 1 0 0
2 0 0 3 2 3 1 0 2 1 0
3 0 0 0 3 2 3 1 0 2 1
4 1 0 0 0 3 2 3 1 0 2
5 2 1 0 0 0 3 2 3 1 0
6 0 2 1 0 0 0 3 2 3 1
7 1 0 2 1 0 0 0 3 2 3
8 3 1 0 2 1 0 0 0 3 2
9 2 3 1 0 2 1 0 0 0 3
⇐⇒
A GDRP(10, 2; 10) of type 4123
(transposed)
0 1 2 3
f1,2,3,6g f4,7g f5,9g f0,8gf2,3,4,7g f5,8g f0,6g f1,9gf3,4,5,8g f6,9g f1,7g f0,2gf4,5,6,9g f0,7g f2,8g f1,3gf0,5,6,7g f1,8g f3,9g f2,4gf1,6,7,8g f2,9g f0,4g f3,5gf2,7,8,9g f0,3g f1,5g f4,6gf0,3,8,9g f1,4g f2,6g f5,7gf0,1,4,9g f2,5g f3,7g f6,8gf0,1,2,5g f3,6g f4,8g f7,9g
J. Yin, Sept. 14, 200617
Japan Seminar Codes From Combinatorial Designs
1.3 CCCs from Frequency Rectangles
• A frequency rectangle of type FR(m,λ) is an m× (λm)
matrix over an m-set S such that every element of S occurs
exactly λ times in each row and once in each column.
• The positive integer λ is called the frequency of elements in the
rectangle. When λ = 1, it is nothing else than Latin square of
order m.
J. Yin, Sept. 14, 200618
Japan Seminar Codes From Combinatorial Designs
• Two frequency rectangles F1 and F2 of type FR(m,λ) over S are
said to be orthogonal if, when F2 is superimposed on F1, every
ordered pair in S × S appears precisely λ times.
• A set of λm− 1 mutually orthogonal frequency rectangles (or
simply MOFRs) of type FR(m,λ) is called complete. When
λ = 1, it is a complete set of m− 1 mutually orthogonal latin
squares of order m.
J. Yin, Sept. 14, 200619
Japan Seminar Codes From Combinatorial Designs
Theorem 1.3 If there exists a complete set of λq − 1 MOFRs of type
FR(q, λ), then so does an optimal
(λq, q(λq − 1), λ(q − 1), [λq]; q)-CCC,
whose size meets the bound in Lemma 1.1.
J. Yin, Sept. 14, 200620
Japan Seminar Codes From Combinatorial Designs
Theorem 1.4 For any prime power q and positive t, there exists a
complete set of qt − 1 MOFRs of type FR(q, qt−1) over GF(q), or
equivalently an optimal
(qt, q(qt − 1), qt − qt−1, [qt−1, qt−1, · · · , qt−1]; q)-CCC.
J. Yin, Sept. 14, 200621
Japan Seminar Codes From Combinatorial Designs
• The Proof of Theorem 1.4
Consider the qt − 1 distinct polynomials over GF(q) of the form
fα, β1,...,βt(x, y1, . . . , yt) = αx + β1y1 + . . . + βtyt,
where α 6= 0, (β1, · · · , βt) 6= (0, . . . , 0) and no two of them are
nonzero multiples of each other.
J. Yin, Sept. 14, 200622
Japan Seminar Codes From Combinatorial Designs
For any such polynomial, we construct a q × qt matrix F overGF(q) so that its rows are indexed by q elements of GF(q) andcolumns are indexed by qt t-tuples over GF(q), and the entry in thecell (a, (b1, . . . , bt)) is fα, β1,··· ,βt(a, b1, . . . , bt). This produces acomplete set of qt − 1 MOFRs of type FR(q, qt−1) over GF(q)
J. Yin, Sept. 14, 200623
Japan Seminar Codes From Combinatorial Designs
1.4 CCCs from Nonlinear Functions
• We use Carlet and Ding (J. Complexity 2004), Coulter
and Matthews (DCC 1997) as our key references on
nonlinear Functions.
• Let (A, +) and (B, +) be two finite abelian group. A function
f : A −→ B is called linear if f(x + y) = f(x) + f(y) for all
x, y ∈ A.
J. Yin, Sept. 14, 200624
Japan Seminar Codes From Combinatorial Designs
• A robust measure of the nonlinearity of a function f : A −→ B
using the derivatives Daf(x) = f(x + a)− f(x) is given by
Pf = max06=a∈A
maxb∈B
|{x ∈ A : Daf(x) = b}||A| ,
where |A| denotes the cardinality of the set A.
J. Yin, Sept. 14, 200625
Japan Seminar Codes From Combinatorial Designs
• For any fixed element a ∈ A, the sets {x ∈ A : Daf(x) = b}(b ∈ B) constitute a partition of A, and hence
|A| =∑
b∈B
|{x ∈ A : Daf(x) = b}| ,
namely,
∑
b∈B
|{x ∈ A : Daf(x) = b}||A| = 1.
J. Yin, Sept. 14, 200626
Japan Seminar Codes From Combinatorial Designs
• The maximum value of the |B| non-negative integers
|{x ∈ A : f(x + a)− f(x) = b}||A| , b ∈ B
is greater than or equal to its mean for any a ∈ A \ {0}. Therefore,
Pf ≥ 1
|B| .
J. Yin, Sept. 14, 200627
Japan Seminar Codes From Combinatorial Designs
• The smaller the value of Pf , the higher the corresponding
nonlinearity of f . If f is linear, then Pf = 1.
• A function f : A −→ B is called a perfect nonlinear function
(PNF) if it has perfect nonlinearity, that is, Pf = 1|B| .
J. Yin, Sept. 14, 200628
Japan Seminar Codes From Combinatorial Designs
• Since the maximum of a sequence of numbers equals its mean if
and only if the sequence is constant, Pf = 1|B| if and only if, for
every b ∈ B and every a ∈ A∗ = A \ {0}, the quantity
|{x ∈ A : Daf(x) = b}| has value |A||B| (this is possible only if
|B| divides |A|).• A perfect nonlinear function f : GF(2)n −→ GF(2) (n > 1) is a
bent function.
J. Yin, Sept. 14, 200629
Japan Seminar Codes From Combinatorial Designs
• A perfect nonlinear function from a finite abelian group to a finite
abelian group of the same order is called a planar function in
finite geometry.
• Planar functions were introduced by Dembowski and Ostrom
(1968) for the construction of affine planes.
• Perfect nonlinear functions introduced by Nyberg (1992) and bent
functions introduced by Rothaus (1976) are extensions of planar
functions.
J. Yin, Sept. 14, 200630
Japan Seminar Codes From Combinatorial Designs
The Encoding Method
Let Π be a PFN from an abelian group (A, +) of order n to an
abelian group (B, +) of order m. Write
A = {a0, a1, · · · , an−1}, B = {b0, b1, · · · , bm−1}.Define
wi = |{x ∈ A : Π(x) = bi}|for each i, and
CΠ = {(Π(a0 + ai), · · · , Π(an−1 + ai)) : 0 ≤ i ≤ n− 1}.
J. Yin, Sept. 14, 200631
Japan Seminar Codes From Combinatorial Designs
• Theorem 1.5 (Ding and Yin (IEE IT 2005-1)) The CΠ defined
above is an (n, n, (m− 1)n/m, [w0, w1, · · · , wm−1; m]) CCC, and is
optimal with respect to the LFVC Bound.
• Applying Theorem 1.5, the known perfect nonlinear functions can
be employed to obtain optimal CCCs, provided that we are able to
determine the frequencies wi. Below is one sample.
J. Yin, Sept. 14, 200632
Japan Seminar Codes From Combinatorial Designs
Example: Let Π(x) =∑t
i=1 x2i−1x2i be the PNF from GF(q)2t to
GF(q). Consider the equation
t∑i=1
z2i−1z2i = a, a ∈ GF(q).
It has Ca solutions of (z1, z2, · · · , z2t) ∈ GF(q)2t. It can be proved that
Ca =
{q2t−1 + (q − 1)qt−1, a = 0
q2t−1 − qt−1, a 6= 0.
J. Yin, Sept. 14, 200633
Japan Seminar Codes From Combinatorial Designs
• It follows that by Theorem 1.5 CΠ defined above is a
(q2t, q2t, (q − 1)q2t−1,
[q2t−1 + (q − 1)qt−1, q2t−1 − qt−1, · · · , q2t−1 − qt−1])
CCC, and is optimal with respect to the LFVC Bound.
J. Yin, Sept. 14, 200634
Japan Seminar Codes From Combinatorial Designs
1.5 CCCs from Partitioned DFs
• Let F = {D1, D2, · · · , Ds} be a collection of subsets (called base
blocks) of an additive abelian group G of order v, and
K = [|D| : D ∈ F ], denoted by an exponential form, is the list of
sizes of base blocks. If the difference list (multiset)
∆F =s⋃
i=1
∆Di =s⋃
i=1
{a− b : a, b ∈ Di and a 6= b}
contains every nonzero element exactly λ times, then F is said to
be a (v, K, λ) difference family, or a (v, K, λ)-DF for short, in G.
J. Yin, Sept. 14, 200635
Japan Seminar Codes From Combinatorial Designs
• The above definition is to say that F is a (v, K, λ)-DF (or PDF)
iff the function
dF(g) :=∑D∈F
|(D + g) ∩D|
takes on a constant value λ when w ranges over all the nonzero
elements of G, where D + g = {x + g : x ∈ D}.• This function is referred to as a difference function defined on
G \ {0}.
J. Yin, Sept. 14, 200636
Japan Seminar Codes From Combinatorial Designs
• In the case that F form a partition of G, we call it partitioned,
or a (v, K, λ)-PDF.
• Theorem 1.6 (Ding and Yin (IEE IT 2005-2))
If a (v, [λ0, λ1, . . . , λq−1], λ)-PDF exists, then so does an optimal
GDRP(n, λ; v) of type {λ0, λ1, · · · , λq−1}, or equivalently an
optimal (n, v, n− λ, [λ0, λ1, . . . , λq−1]; q) CCC (meeting the LFVC
Bound).
J. Yin, Sept. 14, 200637
Japan Seminar Codes From Combinatorial Designs
2.1. Deletion-Correcting Codes
• The conventional error-correcting codes deal with the errors of
symmetric form
x = abccdef −→ y = afcedeb.
Here, Hamming distance d(x,y) = 3.
• During transmission of information strings there exist the errors of
the asymmetric form
abccdef −→ abcdf (2 symbols are deleted)
abccdef −→ abcacdecf (2 symbols are added)
J. Yin, Sept. 14, 200638
Japan Seminar Codes From Combinatorial Designs
• Whenever asymmetric errors occur, the length of the received
string is generally different from the length of the transmitted
string. This makes it impossible to use Hamming metric as a
measure for the distance between two sequences.
• Let x ∈ Qn and y ∈ Qm, where Q stands for an alphabet of size q.
The Levenshtein distance l(x,y) between x and y is defined to
be the smallest number of deletions and insertions needed to
change x to y.
J. Yin, Sept. 14, 200639
Japan Seminar Codes From Combinatorial Designs
• Let Dt(c) denote the set of vectors a ∈ Qn−t that are obtained by
deleting any t components from c. A code C ⊆ Qn is said to be a
t-deletion-correcting code if Dt(c1)⋂
Dt(c2) = ∅ for all
c1, c2 ∈ C with c1 6= c2.
• For any code C ⊆ Qn, define
l(C) = min{l(x,y) : x,y ∈ Qn and x 6= y}.A code C is a t-deletion-correcting code if and only if l(C) > 2t.
J. Yin, Sept. 14, 200640
Japan Seminar Codes From Combinatorial Designs
• A code c is capable of correcting t insertion and/or deletions if
and only if l(C) > 2t. Hence, a t-deletion-correcting code can
correct any combination of up to t deletions and insertions.
• A t-deletion-correcting code code C ⊆ Qn is called perfect, or a
T ∗(n− t, n, q)-code, if Dt(c) (c ∈ C) partition Qn−t.
J. Yin, Sept. 14, 200641
Japan Seminar Codes From Combinatorial Designs
• The possibility of packet loss on internet transmissions has
renewed interest in deletion-correcting codes. The problem is this:
you send n symbols, but only n− t arrive. The t symbol are
deleted, but neither you nor the receiver know which of the n
symbols were lost. One needs to design a code that can correct
such errors.
• D’Yachkov et al. (J. Combin. Optimization 2003) developed a link
between deletion-correcting codes and DNA-Codes.
J. Yin, Sept. 14, 200642
Japan Seminar Codes From Combinatorial Designs
• To my knowledge, very little is known about the theory of
deletion-correcting codes.
• Research on deletion-correcting codes in combinatorial community
has mainly concentrated on the existence problem of a
T ∗(n− t, n, q)-code.
J. Yin, Sept. 14, 200643
Japan Seminar Codes From Combinatorial Designs
The known T ∗(n− t, n, q)-codes
¨ For any positive integer v, a T ∗(2, k, v)-code exists whenever
3 ≤ k ≤ 5 (Levenshtein (DMA 1992); Bourse (DCC 1995);
Mahmoodi (DCC 1998)), where the T ∗(2, 3, v)-codes obtained by
Levenshtein are largest.
¨ For any positive integer v, a T ∗(2, 6, v)-code exists with possible
exceptions of v ∈ {173, 178, 203, 208}(Yin (DCC 2001); Shalaby,
Wang and Yin (DCC 2002)).
J. Yin, Sept. 14, 200644
Japan Seminar Codes From Combinatorial Designs
¨ (Levenshtein (DMA 1992) showed that a T ∗(3, 4, v) -code exists
for all even integers v and left the problem for odd integers open.
Recently, Wang and Ji (JCD 2005) tackled this problem
completely.
¨ A T ∗(2, 7, v)-code exists with a handle of possible exceptions of v
(Wang and Yin (IEEE IT 2006).
J. Yin, Sept. 14, 200645
Japan Seminar Codes From Combinatorial Designs
• The another problem is to find the largest codes. As before, let
A(n, t; q) denote the maximum size of a t-deletion-correcting code
C ⊆ Qn. Such a code with A(n, t; q) codewords is said to be
optimal.
• It is much more difficult generally to obtain upper bounds for
deletion-correcting codes than for conventional error-correcting
codes, since the disjoint code spheres Dt(c) associated with the
codewords do not all have the same size.
J. Yin, Sept. 14, 200646
Japan Seminar Codes From Combinatorial Designs
• The upper bound in the size of a T ∗(2, k, v)-code was given by
Bours (DCC 2005) which we mention later.
• Levenshtein showed that A(n, 1; 2) ∼ 2n
n, as n →∞.
• Besides, very little is known about the values of A(n, t; q). Even
for q = 2 and t = 1, the value of A(n, 1; 2) with n ≥ 9 hasn’t yet
determined.
J. Yin, Sept. 14, 200647
Japan Seminar Codes From Combinatorial Designs
The equivalent definition of T ∗(t, k, v)-codes
• A word (vector) x ∈ Qm is said to be a subword of a word
x ∈ Qn (m ≤ n) if x can be obtained from y by deleting (n−m)
symbols, or equivalently y can be obtained from x by inserting
(n−m) symbols.
• A perfect (k − t)-deletion-correcting code, or a T ∗(t, k, v)-code,
is a subset C ⊆ Qk such that every word of Qt occurs as a
subword in exactly one codeword of C. Here Q is the alphabet of
cardinality v.
J. Yin, Sept. 14, 200648
Japan Seminar Codes From Combinatorial Designs
• From this definition, we see easily that a T ∗(t, k, v)-codes is
capable of correcting any combination of up to (k − t) deletions
and insertions of symbols, since any two distinct codewords in C
cannot share a common subword of length t or longer.
• This definition also suggests a link between T ∗(t, k, v)-codes and
transitive Stainer systems in design theory. If C is a T ∗(2, k, v)-
code, we can verify readily that
|C| ≤v
⌊2(v−1))
k−1
⌋
k
+ v
J. Yin, Sept. 14, 200649
Japan Seminar Codes From Combinatorial Designs
T ∗(2, k, v)-codes from directed BIBD
• A set of k elements {a1, a2, · · · , ak} is said to be transitively
ordered by ai < aj for 1 ≤ i < j ≤ k. In other words, a
transitively ordered k-tuple (a1, a2, · · · , ak) consists of (k − 1)k/2
ordered pairs (ai, aj), 1 ≤ i < j ≤ k.
• A directed (DBIBD) with block size k and order v, denoted by
DB(k, 1; v), is a design (X,A) where A is a set of transitively
ordered k-tuples (blocks) of X, such that every ordered pair of
distinct points of X occurs in exactly one block.
J. Yin, Sept. 14, 200650
Japan Seminar Codes From Combinatorial Designs
• Given a DB(k, 1; v), according to its definition, we may take its
point set as the alphabet Q. Then its blocks form
2v(v − 1)/k(k − 1) codewords of length k, in which every pair of
distinct symbols of Q occurs as a “subword” exactly once, while
any pair of the form (x, x) does not occur. It follows that a
T ∗(2, k, v)-code over Q can be obtained by taking all blocks of the
DB(k, 1; v) and all sequences of length k of the form (x, x, · · · , x),
where x runs over all symbols of Q. Furthermore, we have the
following construction.
J. Yin, Sept. 14, 200651
Japan Seminar Codes From Combinatorial Designs
• Theorem 2.1 (Bours (DCC 1995)) Suppose that k ≥ 3 is an
integer and a DB(k, 1; v) exists. Then a T ∗(2, k, v + 1)-code and a
T ∗(2, k, v − e)-code exist, where 0 ≤ e ≤ 2 if k ≥ 5 and 0 ≤ e ≤ 3
if k ≥ 7.
J. Yin, Sept. 14, 200652
Japan Seminar Codes From Combinatorial Designs
T ∗(2, k, v)-codes from IDBIBDs
• The use of a DB(k, 1; v) in Theorem 5.1 is restricted by its
parameters k and v, which must satisfy 2(v − 1) ≡ 0 (mod k − 1)
and 2v(v − 1) ≡ 0 (mod k(k − 1)). A more general construction
was established using an incomplete DBIBDs (briefly IDBIBDs).
• An IDBIBD, denoted by IDB(k, 1; v, w), is a DB(k, 1; v) (X,A)
missing the blocks of a DB(k, 1; w) (X,B), where Y ⊆ X and
B ⊆ A, and the subdesign, a DB(k, 1; w), is not necessarily to
exist.
J. Yin, Sept. 14, 200653
Japan Seminar Codes From Combinatorial Designs
• Theorem 2.2 (Yin (DCC 2001)) Suppose that k ≥ 3 and an
IDB(k, 1; v, w) exists. Then
i. there exists a T ∗(2, k, v + 1)-code if a T ∗(2, k, w + 1)-code exists;
ii. there exists a T ∗(2, k, v − e)-code if a T ∗(2, k, w − e)-code exists
and 0 ≤ e ≤ min{w, bk−12c}.
• The resulting code in Theorem 2.2 contains a subcode, which is
quite useful in establishing some more combinatorial constructions
for T ∗(2, k, v)-codes.
J. Yin, Sept. 14, 200654
Japan Seminar Codes From Combinatorial Designs
• Note that for the t ≥ 3 case, both the construction of a T ∗(t, k, v)
-code and the determination of the upper bound for its size are
very difficult task. This can be seen from the fact that a perfect
1-deletion-correcting code can be a T ∗(k − 1, k, v)-code.
J. Yin, Sept. 14, 200655
Japan Seminar Codes From Combinatorial Designs
3.1 Signal Sets (Codebooks)
• An (N, K) signal set (or codebook) is a set C = {c0, ..., cN−1} of
N unit norm 1×K complex vector ci, which is called codewords
of the signal set.
• The alphabet of the signal set is the set of different complex values
that the coordinates of all the codewords take.
• The alphabet size is the number of elements in the the alphabet.
• Here and below the codewords are required to be normalized, i.e.,
the norm of every codeword must be 1.
J. Yin, Sept. 14, 200656
Japan Seminar Codes From Combinatorial Designs
• The root-mean-square (RMS) correlation and the maximum
correlation amplitudes of such a signal set are defined as
Irms(C) :=
√√√√ 1
N(N − 1)
∑0≤i,j≤N−1
i 6=j
|〈ci, cj〉|
=
√√√√ 1
N(N − 1)
∑0≤i,j≤N−1
i 6=j
|cicHj |,
Imax(C) := max0≤i<j≤N−1
|cicHj |,
where cHj is the Hermite transpose of the 1×K complex vector cj,
and 〈ci, cj〉 = cicHj is the standard inner product.
J. Yin, Sept. 14, 200657
Japan Seminar Codes From Combinatorial Designs
• Signal sets with best correlation property are desirable in
code-division multiple-access systems (CDMA). In the CDMA
application, a signal set is used to distinguish between the signals
of different users. To this end, the maximum correlation
amplitude Imax of the signal set should be as small as possible.
J. Yin, Sept. 14, 200658
Japan Seminar Codes From Combinatorial Designs
• Direct sequence code division multiple access (DS-CDMA) is one
of the approaches to spread spectrum for signal transmission.
Optimal and almost optimal (N,K) signal sets with N > K are
desirable in synchronous DS-CDMA systems for reducing
interference, where the number of users N is greater than the
signal space dimension or the spreading factor K. Such signal sets
could be quite useful in 3G and 4G networks for increasing the
subscriber capacity within the limited spectral resource.
J. Yin, Sept. 14, 200659
Japan Seminar Codes From Combinatorial Designs
• Welch’s bounds (Welch (IEEE IT 1974)) For any (N, K)
signal set C with N ≥ K,
Irms(C) ≥√
N −K
(N − 1)K, (1)
with equality if and only if∑N
i=0 cHi ci = (N/K)IK , where IK
denotes the K ×K identity matrix.
J. Yin, Sept. 14, 200660
Japan Seminar Codes From Combinatorial Designs
• We have also
Imax(C) ≥√
N −K
(N − 1)K, (2)
with equality if and only if for all pairs (i, j) with i 6= j
|cicHj | =
√N −K
(N − 1)K. (3)
J. Yin, Sept. 14, 200661
Japan Seminar Codes From Combinatorial Designs
• If the equality holds in (1), then C is referred to as a
Welch-bound-equality (WBE) signal set.
• A signal set meeting the bound of (2) is called a
maximum-Welch-bound-equality (MWBE) signal set.
• An MWBE signal set must be a WBE signal set, but a WBE
signal set may not be an MWBE signal set. MWBE signal sets
form a subset of WBE signal sets.
J. Yin, Sept. 14, 200662
Japan Seminar Codes From Combinatorial Designs
• It is fairly easy to construct WBE signal sets. Every linear error
correcting code whose dual code with Hamming distance at least 3
yields a WBE signal set (Massey and Mittelholzer (1993); Sarwate
(1998)).
• However, MWBE signal sets are very hard to construct, as
pointed out by Sarwate (1998).
J. Yin, Sept. 14, 200663
Japan Seminar Codes From Combinatorial Designs
• The known classes of MWBE signal sets:
¨ The trivial (N, N) and (N, N − 1) MWBE signal sets;
¨ The (N, K) MWBE signal sets based on conference matrices
(Conwayet al. (1996); Strohmer and Heath (2003)), when
N = 2K = 2d+1 and N = 2K = pd + 1 with p a prime number,
where d is a positive integer.
¨ (N, K) MWBE signal sets based on (N, K, λ)-DSs which we will
present later.
J. Yin, Sept. 14, 200664
Japan Seminar Codes From Combinatorial Designs
• It was shown in (Strohmer and Heath (2003)) that
♦ No (N,K) real signal set C can meet the Welch bound of (2), if
N > K(K + 1)/2.
♦ No (N,K) signal set C can meet the Welch bound of (2), if
N > K2.
• From the above fact, we see that Welch’s bound on the maximum
correlation amplitude cannot be achieved in certain cases either,
as it is not tight in these cases.
J. Yin, Sept. 14, 200665
Japan Seminar Codes From Combinatorial Designs
• The following bounds developed by Levenstein (Kabatyanskii and
Levenstein (1978); Levenstein (1983)) are better than Welch’s
bound in certain cases.
¨ For any (N,K) real signal set C with N > K(K + 1)/2, we have
Imax(C) ≥√
3N −K2 − 2K
(K + 2)(N −K). (4)
¨ For any (N, K) complex signal set C with N > K, we have
Imax(C) ≥√
2N −K2 −K
(K + 1)(N −K). (5)
J. Yin, Sept. 14, 200666
Japan Seminar Codes From Combinatorial Designs
• Another lower bound documented in (Xia et al. (IEEE IT 2005) is
the following:
Imax(C) ≥ 1− 2N− 1K−1 . (6)
J. Yin, Sept. 14, 200667
Japan Seminar Codes From Combinatorial Designs
• If N < K(K + 1)/2 (respectively N < K2), the Welch bound on
real (respectively, complex) signal sets is the best among the three
bounds. However, the Levenstein bound of (4) on real signal sets
is tighter than Welch’s bound if N > K(K + 1)/2, and that the
Levenstein bound of (5) on complex signal sets is tighter than
Welch’s bound if N > K2. The bound of (6) could be negative,
e.g., when N = 71 and K = 35, and thus makes no sense in
certain cases. But in some cases the lower bound of (6) could be
tighter, compared with the Welch and Levenstein bounds.
J. Yin, Sept. 14, 200668
Japan Seminar Codes From Combinatorial Designs
3.2 Signal Sets from Cyclic DSs
• Define ψN(n) = e2nπ√−1/N . Then ψN is a group character of
(ZN , +). For any K-subset D = {d1, d2, · · · , dK} of ZN , define a
signal set given by
CD = {ci : i ∈ ZN}. (7)
where for each i with 0 ≤ i ≤ N − 1
ci :=1√K
(ψN(id1), ψN(id2), · · · , ψN(idk)) . (8)
J. Yin, Sept. 14, 200669
Japan Seminar Codes From Combinatorial Designs
• Theorem 3.1 (Xia et al. (IEEE IT 2005)) CD is an (N, K)
MWBE signal set if and only if D is a cyclic (N, K, λ)-DS in ZN .
• Theorem 3.1 works only for cyclic difference sets. It is a
generalization of a specific family of equiangular tight frames
(another name for siginal sets) obtained by Konig (Konig 1979).
• Theorem 3.1 produces a number of MWBE signal sets based on
known cyclic difference sets.
J. Yin, Sept. 14, 200670
Japan Seminar Codes From Combinatorial Designs
3.3 Signal Sets from DSs in GF(q)
• Let x0, x1, · · · , xq−1 denote all the elements of the finite field
GF(q) with q = pm. Let εp = e2π√−1/p and Tr(x) be the absolute
trace function from GF(q) to GF(p). For any K-subset
D = {d1, d2, · · · , dK} of GF(q), define a signal set given by
CD = {ci : i = 0, 1, · · · , q − 1}. (9)
where for each i with 0 ≤ i ≤ q − 1
ci :=1√K
(εTr(xid1p , εTr(xid2
p , · · · , εTr(xidKp ,
). (10)
J. Yin, Sept. 14, 200671
Japan Seminar Codes From Combinatorial Designs
• Theorem 3.2 (Ding (IEE IT 2006)) CD is an (N, K) MWBE
signal set if D is a (N,K, λ)-DS in (GF(q), +) where K > 1 .
• Theorem 3.2 works both cyclic and noncyclic difference sets in
GF(q).
J. Yin, Sept. 14, 200672
Japan Seminar Codes From Combinatorial Designs
3.4 Signal sets from PNFs
• Assume that m is a positive integer and p is an odd prime and
q = pm so that there are planar functions (GF(q), +) to
(GF(q), +). As before, use x0, x1, · · · , xq−1 to denote all the
elements of the finite field GF(q). For any positive integer n,
define εn = e2π√−1/n. Let Tr denote the absolute trace function on
GF(q). Define
ψ(x) := εTr(x)p . (11)
Then ψ is an additive group character of GF(q).
J. Yin, Sept. 14, 200673
Japan Seminar Codes From Combinatorial Designs
• Let Π be a perfect nonlinear function from (GF(q), +) to
(GF(q), +). For each pair (a, b) ∈ GF(q)2, define the unit-norm
vector
c(a,b) =1√q
(ψ(aΠ(x0) + bx0), ψ(aΠ(x1) + bx1), ..., ψ(aΠ(xq−1) + bxq−1)) .
Then form a signal set
CΠ = {c(a,b) : (a, b) ∈ GF(q)2}⋃
Eq. (12)
J. Yin, Sept. 14, 200674
Japan Seminar Codes From Combinatorial Designs
• Here, Eq is the standard orthogonal basis of the q-dimensional
Hilbert space, consisting of the following vectors:
e(q)1 = (1, 0, 0, 0, · · · , 0, 0),
e(q)2 = (0, 1, 0, 0, · · · , 0, 0),
......
...
e(q)q = (0, 0, 0, 0, · · · , 0, 1),
J. Yin, Sept. 14, 200675
Japan Seminar Codes From Combinatorial Designs
• Theorem 3.3 (Ding and Yin (IEEE Commun. 2006) CΠ is a
(q2 + q, q) signal set with Imax(CΠ) = 1√q.
J. Yin, Sept. 14, 200676
Japan Seminar Codes From Combinatorial Designs
4.1 Optimal OOCs
• A (v, k, λ1, λ2) optical orthogonal code (briefly
(v, k, λ1, λ2)-OOC), C, is a family of (0, 1)-sequences (codewords)
of length v and weight k satisfying the following two properties:
¨∑
0≤t≤v−1xtxt+i ≤ λ1 for any x = (x0, x1, . . . , xv−1) ∈ C and any
integer i 6≡ 0 mod v (the auto-correlation property) ;
¨∑
0≤t≤v−1xtyt+i ≤ λ2 for any x = (x0, x1, . . . , xv−1) ∈ C,
y = (y0, y1, . . . , yv−1) ∈ C with x 6= y, and any integer i (the
cross-correlation property) .
J. Yin, Sept. 14, 200677
Japan Seminar Codes From Combinatorial Designs
• All subscripts here are reduced modulo v so that only periodic
correlations are considered.
• OOCs are applied in multimedia transmission in fiber-optic LANs
and in multirate fiber-optic CDMA systems which require binary
sequences with good correlation properties.
• Research on (v, k, λ1, λ2)-OOCs in combinatorial community has
mainly concentrated on the case where λ1 = λ2 = 1. In this case,
the CCC is called a (v, k, 1)-OOC, and it is optimal if it contains
b v−1k(k−1)
c codewords.
J. Yin, Sept. 14, 200678
Japan Seminar Codes From Combinatorial Designs
4.2 The Comb. Characterization
• Let F be a family of t k-subsets of Zv. If there exists a (minimum)
constant value λ such that dF(g) ≤ λ for any nonzero residue in
Zv, then F is well known as a cyclic difference packing, or a
(v, k, λ)-CDP in short. When b v−1k(k−1)
c then it is optimal.
• Theorem 4.1 (See, for example, Yin (DM 185 (1998), PP. 201))
An optimal (v, k, 1)-OOC is equivalent to an optimal (v, k, 1)-CDP.
J. Yin, Sept. 14, 200679
Japan Seminar Codes From Combinatorial Designs
• The construction of a (v, k, 1)-CDP with b v−1k(k−1)
c base blocks is a
rather difficult task. The existence of a (v, 4, 1)-CDP with bv−112c
base blocks is far from complete to this day. For the recent results,
the reader may refer to the new version of The CRC Handbook
of Combinatorial Designs
(http://www.cems.uvm.edu/ jcd/hcdproofread/). Its publication
is due this year.
J. Yin, Sept. 14, 200680
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5. Splitting A-Codes and EDFs
• Consider a collection F = {D1, D2, · · · , Du} of subsets (base
blocks) of an additive abelian group G of order v. If the base
blocks in F are mutually disjoint, then we have another
difference function, called external difference function, that is
defined as
edF(g) :=∑
1≤i6=j≤u
|(Di + g) ∩Dj|,
for g ∈ G \ {0}.
J. Yin, Sept. 14, 200681
Japan Seminar Codes From Combinatorial Designs
• If edF(g) takes on a constant value λ when g ranges over all the
nonzero elements of G. Then F is referred to as an external
difference family (EDF), or a (v, {k1, k2, · · · , ku}, λ)-EDF in
short. In the case |Di| = k for any i (1 ≤ i ≤ u), it is termed a
(v, k, λ; u)-EDF.
• The notion of an (n, k, λ; u)-EDF of index λ = 1
was first defined by Ogata et al. (DM 2004).
J. Yin, Sept. 14, 200682
Japan Seminar Codes From Combinatorial Designs
• Using such an EDF, Ogata et al. constructed an A-code where the
plaintext (source state) space S = {1, 2, · · · , u}, message space Mand key space E are both taken to be G. For i ∈ S and e ∈ E ,
e(i) = Di. The term “splitting” means that a message is not
uniquely determined by the plaintext and the key, that is very
important in the context of authentication with arbitration. Here
|Di| = k > 1. So the resulting A-code is k-splitting. They also
gave a way to obtain (k, n) secret scheme against cheaters from a
EDFs, and proved that the scheme from such an EDF is optimal
with respect a certain bound.
J. Yin, Sept. 14, 200683
Japan Seminar Codes From Combinatorial Designs
6.1 Comma Free Codes
• The code synchronization problem
In a comma free code, the data stream consists of consecutive
messages each being a sequence of n consecutive symbols. The
synchronization problem that arises at the receiving end is the task
to partition correctly the data stream into messages of length n.
J. Yin, Sept. 14, 200684
Japan Seminar Codes From Combinatorial Designs
Correct decoding:
· · · ︷ ︸︸ ︷a1a2 · · · an
︷ ︸︸ ︷b1b2 · · · bn · · ·
Incorrect decoding:
︷ ︸︸ ︷· · · a1 · · · ai
︷ ︸︸ ︷ai+1 · · · anb1 · · · bi
︷ ︸︸ ︷· · · bn · · ·
It is the concatenation of the end of one message with the beginningof another message, denoted by Ei(a,b).
J. Yin, Sept. 14, 200685
Japan Seminar Codes From Combinatorial Designs
• One way to resolve the synchronization problem is by requiring
that no codeword c coincides with the concatenation Ei(a,b),
where a and b are two codewords (not necessarily distinct).
• Let (Zq, C) be a (n,M, d; q) code. The comma-free index ρ(C)
of the code C is defined as
ρ(C) = min d(c, Ei(a,b))
where the minimum is taken over all the codewords a,b, c ∈ C
and all i = 1, 2, · · · , n− 1, and d(c, Ei(a,b)) stands for the
Hamming distance.
J. Yin, Sept. 14, 200686
Japan Seminar Codes From Combinatorial Designs
• The comma-free index ρ(C) allows one to distinguish a codeword
from a concatenation of two codewords even in case that up to
b(ρ(C)− 1)/2c errors have occurred (and hence allows
synchronization)(see Golomb et al., 1958).
J. Yin, Sept. 14, 200687
Japan Seminar Codes From Combinatorial Designs
6.2 Comma free codes from DSSs
• Consider a collection F = {D1, D2, · · · , Du} of u disjoint subsets
(base blocks) of Zv. If there exists a (maximum) constant value λ
such that edF(g) ≥ λ for any nonzero residue (mod v). Then F is
called difference system of sets (DSSs) or a
(v, {k1, k2, · · · , ku}, λ)-DSS.
• When |Di| = k for any i (1 ≤ i ≤ u), the DSS is called regular.
As with EDF, we simply say that F is a (v, k, λ; u)-DSS in this
case.
J. Yin, Sept. 14, 200688
Japan Seminar Codes From Combinatorial Designs
• A perfect (v, {k1, k2, · · · , ku}, λ)-DSS is a cyclic EDF.
• Levenshtein (1971, 2004) gave a method to construct codes with
prescribed comma-free index by way of a DSS.
• The detailed can be found in a recent survey paper by Tonchev
(Finite Fields and Their Applications 11 (2005), 601–621).
J. Yin, Sept. 14, 200689
Japan Seminar Codes From Combinatorial Designs
• The application of a DSS of index λ {Dj : j = 0, 1, · · · , q − 1} in
Zn to code synchronization requires the redundancy
rq(n, λ) =
q−1∑j=0
|Dj|
is as small as possible. A DSS is optimal if its redundancy is
equal to rq(n, λ).
J. Yin, Sept. 14, 200690
Japan Seminar Codes From Combinatorial Designs
• Levenshtein (1971) proved that
rq(n, λ) ≥√
λq(n− 1)q − 1
(13)
with the equality if and only if the DSS is perfect and regular.
J. Yin, Sept. 14, 200691
Japan Seminar Codes From Combinatorial Designs
Appendix
The Link between PDFs and EDFs
• PDFs, EDFs and DSSs we have touched upon are different
combinatorial objects, but are closely related. A perfect DSS is
a cyclic EDF. A convenient way to mention the link between
PDFs and EDFs is to use a group ring. Let G be an additive
abelian group and Z the ring of integers. Define the ring of the
formal polynomials over Z of an indeterminate x
Z[x] =
{∑g∈G
agxg : ag ∈ G
}.
J. Yin, Sept. 14, 200692
Japan Seminar Codes From Combinatorial Designs
• The addition of Z[x] is given by∑g∈G
agxg +
∑g∈G
bgxg =
∑g∈G
(ag + bg)xg
• The multiplication of Z[x] is given by(∑
g∈G
agxg
)(∑g∈G
bgxg
)=
∑
h∈G
(∑g∈G
agbh−g
)xh.
J. Yin, Sept. 14, 200693
Japan Seminar Codes From Combinatorial Designs
• The additive unity and the multiplicative unity of Z[x] are defined
respectively as
0 :=∑g∈G
0xg and 1 := x0.
• The Hall polynomial∑
g∈S xg (S ⊆ G) is often simply denoted by
S(x).
J. Yin, Sept. 14, 200694
Japan Seminar Codes From Combinatorial Designs
• Now let F = {D1, D2, · · · , Du} is a partition of an abelian group
G of order v. With the above observations, we can see
¨ F is a (v, {k1, k2, · · · , ku}, λ)-PDF in G provided that
u∑i=1
Di(x)Di(x−1) = v − λ + λG(x).
¨ F is a (v, {k1, k2, · · · , ku}, λ)-EDF in G provided that∑
1≤i 6=j≤u
Di(x)Dj(x−1) = −λ + λG(x).
J. Yin, Sept. 14, 200695
Japan Seminar Codes From Combinatorial Designs
• In summary, F is a (v, {k1, k2, · · · , ku}, λ)-PDF in G if and
only if F is a (v, {k1, k2, · · · , ku}, v − λ)-EDF in G
• A similar relationship between disjoing DFs and external DFs was
mentioned in (Chang and Ding, 2006).
• A DSS is essentially a cyclic external difference covering. We can
establish the link between DSSs and CDPs in a similar vein.
J. Yin, Sept. 14, 200696
Japan Seminar Codes From Combinatorial Designs
THANKS!