Kite Codes over Groups

Xiao Ma∗, Shancheng Zhao∗, Kai Zhang∗, and Baoming Bai†

∗School of ECE, Sun Yat-sen University

Guangzhou 510006, China†State Key Lab. of ISN, Xidian University

Xi’ an 710071, China

Abstract—Kite codes, which were originally defined over thebinary field, are generalized to arbitrary abelian groups in thispaper. Kite codes are a special class of prefix rateless codes overgroups, which can generate potentially infinite (or as many asrequired) random-like parity-check symbols. In this paper, weconsider four kinds of Kite codes, which are binary Kite codes,Kite codes over one-dimensional lattices, Kite codes over M-PSK signal constellations and Kite codes over multi-dimensionallattices. It is shown by simulations that the proposed codesperform well over additive white Gaussian noise channels.

Index Terms—Adaptive coded modulation, codes over groups,group codes, lattice codes, LDPC codes, RA codes, rate-compatible codes, rateless coding, Raptor codes.

I. INTRODUCTION

Binary linear rateless coding is an encoding method that

can generate potentially infinite parity-check bits for any

given fixed-length binary sequence. Fountain codes constitute

a class of rateless codes, which were first mentioned in [1].

The first practical realizations of Fountain codes were LT-

codes invented by Luby [2]. LT-codes are linear rateless

codes that transform k information bits into infinite coded

bits. LT-codes have encoding complexity of order log k per

information bit on average. The complexity is further reduced

by Shokrollahi using Raptor codes [3]. For relations among

random linear Fountain codes, LT-codes and Raptor codes

and their applications over erasure channels, we refer the

reader to [4]. In addition to their success in erasure channels,

Fountain codes have also been applied to other binary mem-

oryless symmetric channels [5][6][7]. However, it has been

proved [6] that no universal Raptor codes exist for binary input

memoryless symmetric channels (BIMSCs) other than erasure

channels. Recently, a new class of binary rateless codes, called

Kite codes, has been proposed for additive white Gaussian

noise (AWGN) channels [8]. Simulation results have shown

that Kite codes perform well in a wide range from decoding

rate 0.1 to 0.9.

In this paper, we generalize the original Kite codes from the

binary field to arbitrary abelian groups. Kite codes are a special

class of prefix rateless forward error correction (FEC) codes

over arbitrary abelian groups, which can generate as many

as required random-like parity-check symbols. A Kite code is

specified by its dimensions, a real sequence and an abelian

group along with a subset of bijective transformations over it.

By picking tailored parameters, Kite codes can be constructed

as low-density parity-check (LDPC) codes, group codes [9]

and lattice codes [10]. This makes them more attractive for

certain applications.

II. KITE CODES OVER ABELIAN GROUPS

A. Finite Abelian Groups and Their Transformations

A finite abelian group (A,+) is defined as a finite set Atogether with a binary operation + : A × A 7→ A satisfying

the following conditions [11].

1) There exists an element θ ∈ A such that α + θ = αholds for any α ∈ A. It can be proved that θ is unique.

We call θ the identity element of A and denote it by 0for convenience.

2) For any α, β, γ ∈ A, (α + β) + γ = α + (β + γ).3) For any α, β ∈ A, α + β = β + α.

4) For any α ∈ A, there exists an element β ∈ A such

that α + β = 0. It can be proved that β is uniquely

determined by α. We call such a β the negative element

of α and denote it by −α for convenience.

A transformation f : A 7→ A assigns to each α ∈ A exactly

one element β ∈ A. The collection of all transformations

defined over A is denoted by T . The collection of all bijective

transformation is denoted by S. A transformation f is called an

endomorphism of A if f(α+β) = f(α)+f(β) for all α, β ∈A. An endomorphism f is also called an automorphism if it

is bijective. The collection of all endomorphisms is denoted

by End(A). We use 0A to denote the zero transformation that

assigns to each α ∈ A the identity element, and 1A to denote

the identical transformation that transforms each α ∈ A to

itself.

We give some examples that are commonly used in the

coding theory and practice.

Examples.

1) Consider the additive group Fq of the finite field of size

q. For any nonzero α ∈ Fq , we can define a bijective

transformation β 7→ αβ, β ∈ Fq .

2) Consider the additive group of the modulo ring Zm. For

any α which is coprime with m, we can define a bijective

transformation β 7→ αβ, β ∈ Zm.

3) Consider the M-ary phase-shift keying (M-PSK) signal

constellation X =

exp(

2kπ√−1

M

)

, 0 ≤ k < M

. For

any α ∈ X , we can define a bijective transformation

β 7→ αβ, β ∈ X .

4) Consider a quotient lattice Λ/Λ0, where Λ ⊂ Rm is a

lattice with Λ0 as a sub-lattice. For any α ∈ Λ/Λ0, we

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Buffer of size k for information sequence

At time t, generate a random variable ;

randomly select summands from the buffer; for each

selected summands, find its image under a random

transformation in ; generate the parity-check sum.

PRNG

w

t-1

v

w

v

v -- systematic information sequence of length k

w -- parity-check sequence of length r = n - k

B(k; p

t

) -- binomial distribution with k Bernoulli trials and success probability p

t

PRNG -- pseudo-random number generator

);(~

tt

pkBθ

t

θ

s

t

Ω

Fig. 1. A systematic encoding algorithm for the proposed Kite code.

can define a bijective transformation β 7→ α + β, β ∈Λ/Λ0.

B. Systematic Rateless Codes Over Groups

Let Aq be a finite abelian group of size q. Let A∞q be the

set of all infinite sequences over Aq . A systematic rateless

code with degree of freedom k is defined as

Cq[∞, k]∆=

c = (v, w) ∈ A∞q | v ∈ Ak

q

, (1)

which is a finite subset of A∞q with size qk. Similar to

linear codes, for c = (v, w) ∈ Cq[∞, k], we call c, v and

w coded sequence, information sequence and parity-check

sequence, respectively. With a little bit of confusion, the degree

of freedom k is also referred to as the dimension of the

code Cq[∞, k]. For n ≥ k, we use Cq[n, k] to denote the

prefix code with length n of Cq[∞, k]. For the prefix code,

the coding rate is k/n. If the information sequence v is

drawn from a memoryless source with entropy H(V ), the

information rate is then kH(V )/n bits/symbol. In particular,

if the source is uniformly distributed over Aq , the information

rate is (k log2 q)/n bits/symbol.

We now propose a special class of rateless codes,

called Kite codes. An ensemble of Kite codes, denoted by

K[∞, k; p,Aq,Ω], is specified by its dimension k, a real

sequence p = (p0, p1, · · · , pt, · · ·) with 0 < pt < 1 for t ≥ 0and a subset Ω of bijective transformations over Aq. The real

sequence p is called p-sequence for convenience. The encoder

of a Kite code consists of a buffer of size k, a pseudo-random

number generator (PRNG) and an accumulator, as shown in

Fig. 1.

Let v = (v0, v1, · · · , vk−1) be the information sequence to

be encoded. The corresponding codeword is written as c =(v, w), where w = (w0, w1, · · · , wt, · · ·) is the parity-check

sequence. The task of the encoder is to compute wt for any

t ≥ 0. The encoding algorithm is described as follows.

Algorithm 1: (Recursive Encoding Algorithm for Kite

Codes)

1) Initialization: Initially, set w−1 = 0. Equivalently, set

the initial state of the accumulator to be zero.

2) Recursion: For t = 0, 1, · · · , T − 1 with T as large

as required, compute wt recursively according to the

following procedures.

Step 2.1: At time t ≥ 0, sampling k independent

identically distributed (i.i.d.) random transformations

Ht = (Ht,0,Ht,1, · · · ,Ht,k−1), where

PrHt,i = ht,i =

pt/|Ω|, ht,i ∈ Ω ⊆ S1 − pt, ht,i = 0A

for 0 ≤ i < k;

Step 2.2: Compute the t-th parity-check symbol by wt =wt−1 + st, where st =

∑

0≤i<k ht,i(vi).

Remark. If Ω consists of automorphisms of Aq , the result-

ing codes are group codes [9].

C. Normal Graphical Realizations of Kite Codes

The prefix Kite code K[n, k; p,Aq, Ω] can be represented

by a normal graph [12], which is described as follows and

also shown in Fig. 2.

1) There are in total k information variable nodes, repre-

sented by =©, each of which corresponds to a component

of v.

2) There are in total r∆= n − k check variable nodes,

represented by = , each of which corresponds to a

component of w.

3) Check variable nodes are connected to information vari-

able nodes via two types of nodes in between, which,

represented by by h© and + , are called transformation

nodes and check nodes, respectively.

For decoding, the receiver can perform the sum-product

algorithm (SPA) over the Normal graph [12][13] according

to the following schedules, where the messages processed at

nodes and passed along edges can be in principle probability

mass functions (pmf) of the corresponding random variables.

Algorithm 2: (Decoding Algorithm for Kite Codes: A

Schedule)

1) Initialization: Initially, the messages for information

variable nodes and check variable nodes are assumed

to be computable from the channel observations. These

messages are passed to check nodes. Note that the

messages received at the check nodes from the in-

formation variable nodes need to be permuted by the

transformation nodes.

2) Iterations: One iteration includes two phases as follows.

Step 2.1: The check node takes as input the messages

from the transformation nodes and check variable nodes

and computes the extrinsic messages which are delivered

(permuted if required) to information variable nodes and

check variable nodes;

Step 2.2: The information variable nodes and check vari-

able nodes take the (possibly permuted) messages from

check nodes and deliver as outputs extrinsic messages

to the check nodes.

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Information variable node: k in total

Parity-check variable node: r in total

Check node: r in total

Transformation node

h

h

Fig. 2. A normal graph of a Kite code.

D. Design of Kite Codes and Their Improvements

Given k, a finite abelian group Aq and a subset Ω of

bijective transformations, the performance of a Kite code is

mainly affected by the p-sequence. The task to optimize a Kite

code is to select the whole p-sequence such that all the prefix

codes are good enough. This is a multi-objective optimization

problem and could be very complicated. For simplicity, we

partition the p-sequence into nine segments, each of which

corresponds to a parameter qi, 1 ≤ i ≤ 9. Specifically, for

t ≥ 0, pt is set to be qi if the coding rate k/(k + t + 1) falls

into the interval [i/10, (i+1)/10). Then designing a Kite code

is equivalent to selecting the parameters q = (q9, q8, · · · , q1).This can be done by a greedy optimization algorithm [8] based

on simulation or density evolution [14]. Firstly, we choose q9

such that the prefix code of rate 0.9 is as good as possible.

Secondly, we choose q8 with fixed q9 such that the prefix

code of rate 0.8 is as good as possible. Thirdly, we choose

q7 with fixed (q9, q8) such that the prefix code of rate 0.7is as good as possible. This process continues until q1 is

selected. A construction example of Kite code defined over

F2 for k = 1890 has been given in [8]. For convenience, the

parameters are reproduced in Table I.

Given the parameters q, a randomly generated Kite code

may have information variable nodes with extremely low

degrees (even zero degrees for high-rate codes). This typically

-6 -4 -2 0 2 4 6 8 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR(dB)

R

a

t

e

s

(

b

i

t

s

/

B

P

S

K

)

simulation results

channel capacity

Fig. 3. Performance of the binary Kite code with k = 1890.

causes higher error-floors for Kite codes with high coding

rates. To improve the performance near the error-floor region,

we can modify a randomly generated Kite code by adding

edges between low-degree information variable nodes and low-

degree check nodes.

III. EXAMPLES OF KITE CODES

In this section, we will give some specific examples of

Kite codes and their performances over AWGN channels.

For numerical results, we take the K[n, 1890; p,Aq, Ω] as

examples.

A. Binary Kite Codes

Consider the Kite codes over F2 with Ω = 1A. They are

prefix binary LDPC codes.

Example 1: Consider K[n, 1890; p, F2, 1A]. The p-

sequence is shown in Table I. The signal-to-noise ratio (SNR)

required to achieve bit-error rate (BER) around 10−4 for

different coding rate are shown in Fig. 3 under the assumptions

of AWGN channels, BPSK modulation and SPA decoding al-

gorithm. The decoding algorithm is carried out with maximum

iteration Imax = 50. Also shown in Fig. 3 is the channel

capacity curve. It can be seen from Fig. 3 that the gaps between

the coding rates and the capacities are around 0.15 bits/BPSK

in the SNR range of −5.0 ∼ 7.0 dB. The gaps can be further

narrowed as shown in [8] by increasing the data length k. To

the best of our knowledge, no simulation results are reported in

the literature to illustrate that one coding method can produce

good codes in such a wide range.

B. Kite Codes over One-Dimensional Lattice

Consider the Kite codes over A = Z/mZ, referred to as

Z-Kite codes. Generally, we choose the lattice point in a

coset with least Euclidean distance to the origin as its coset

representative.

Example 2: Consider K[2100, 1890; p, Z/3Z, Ω], where

Z/3Z = −1, 0,+1 and Ω the set of bijective transfor-

mations over Z/3Z. Different from conventional settings for

channel coding, we allow the information sequence v to have

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TABLE ITHE p-SEQUENCE OF KITE CODES WITH k=1890.

t [0,210) [210,472) [472,810) [810,1260) [1260,1890) [1890,2835) [2835,4410) [4410,7560) [7560,17010)

pt

q9 q8 q7 q6 q5 q4 q3 q2 q1

0.0249 0.0072 0.0045 0.0034 0.0021 0.0016 0.0010 0.0006 0.0004

TABLE IISOURCE ENTROPIES AND THEIR DISTRIBUTIONS.

H(V )(bits/symbol) Pr(−1) Pr(0) Pr(+1)log2(3) 0.3333 0.3334 0.33331.3333 0.1886 0.6228 0.18861.2222 0.1595 0.6810 0.15951.0000 0.1135 0.7730 0.11350.8889 0.0946 0.8108 0.0946

non-uniform a priori probability, which may yield shaping

gain [15][16]. As an example, if the entropy of the information

symbol is equal to 1.0 bits/symbol, we have information

rate 0.9 bits/channel symbol. The source distributions and

their entropies H(V ) considered in this paper are shown in

Table II. The symbol error rates (SER) are shown in Fig. 4

when decoded with SPA. It is worth pointing out that the

a priori probabilities are incorporated into the initialization

of the decoding algorithm for information variable nodes but

not for check variable nodes. This is because, under random

bijective transformations, the check variable nodes are usually

uniformly distributed over Z/3Z. In addition, the a priori

probabilities have also been taken into account in the process

of evaluating transmission power given that the average (per

channel symbol) transmission power allocation are the same

for information sequence and parity-check sequence. The

decoding algorithm is carried out with maximum iteration

Imax = 50. It can be seen that, at information rate 0.9

bits/channel symbol and SER = 10−4, this Z-Kite code has

a gain over the binary Kite code in Sec. III-A about 0.6 dB.

Note that this is a rough comparison given that one is SER

and the other is BER.

C. Kite Codes over M-PSK Constellation

Consider the Kite codes over AM =

exp(

2kπ√−1

M

)

, 0 ≤ k < M

, referred to as MPSK-Kite

codes here.

Example 3: Consider K[2100, 1890; p,A8,Ω], where Ω is

the set of bijective transformations over A8. The BER is shown

in Fig. 5 when decoded with SPA. Also shown in Fig. 5 is the

performance of a modified code of K[2100, 1890; p,A8, Ω].The decoding algorithm is carried out with maximum iteration

Imax = 50. It can be seen that the modified code reveals a

lower error floor.

D. Kite Codes over 4-Dimensional Checkerboard Lattice

Consider the Kite codes over A = Λ/Λ0 [17], where Λis a m-dimensional lattice in Euclidean space and Λ0 is a

sub-lattice of Λ. It is referred to as Lattice-Kite codes here.

The coded sequence of Lattice-Kite codes is transformed into

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5

10

-7

10

-6

10

-5

10

-4

10

-3

10

-2

10

-1

Eb/No(dB)

S

y

m

b

o

l

E

r

r

o

r

R

a

t

e

[2100, 1890] Z-Kite code

Fig. 4. Performances of Kite codes [2100, 1890] over Z/3Z with differentsource distributions. From left to right, the curves correspond to informationrates 0.80, 0.90, 1.10, 1.20 and 1.43 bits/channel symbol, respectively.

6 6.5 7 7.5 8 8.5 9

10

-8

10

-7

10

-6

10

-5

10

-4

10

-3

10

-2

10

-1

Eb/No(dB)

S

y

m

b

o

l

E

r

r

o

r

R

a

t

e

8PSK-Kite Code: 2.7 bits/channel symbol

Modified 8PSK-Kite Code: 2.7 bits/channel symbol

Fig. 5. Performances of Kite code [2100, 1890] over 8-PSK.

signal point sequence by mapping each component (coset) to

a lattice point in it with least energy.

Example 4: Consider the 4-D Lattice-Kite code

K[2100, 1890; p, D4/4D4, Ω1] and 2-D Lattice-Kite code

K[2100, 1890; p, Z2/4Z2,Ω2], where Ω1 and Ω2 are the

set of bijective transformations of D4/4D4 and Z2/4Z2,

respectively. Note that Z2/4Z2 is equivalent to 16QAM

signal constellation. The SERs are shown in Fig. 6 when

decoded with SPA. It can be seen that the coding gain of

4-D Lattice-Kite code over 2-D Lattice-Kite code (16QAM)

is about 1.5 dB at error rate 10−5. We observed that the

4-D Lattice-Kite code converges very fast. The decoding

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5 6 7 8 9 10 11

10

-6

10

-5

10

-4

10

-3

10

-2

10

-1

10

0

Eb/No(dB)

S

y

m

b

o

l

E

r

r

o

r

R

a

t

e

2-D Lattice-Kite code: 3.6 bits/2-dimension

4-D Lattice-Kite code: 3.6 bits/2-dimension

Shannon Limit of 16QAM

Fig. 6. Performances of Kite code [2100, 1890] over D4/4D4 and Z2/4Z2.

algorithm for 4-D Lattice-Kite code is carried out with

maximum iteration Imax = 5, whereas Imax = 50 for 2-D

Lattice-Kite code.

IV. CONCLUSION

In this paper, we have proposed a new class of rateless

forward error correction codes, called Kite codes, over arbi-

trary abelian groups which can be applied to AWGN channels.

The Kite codes can generate potentially infinite parity-check

bits with linear complexity. Four kinds of Kite codes, which

are binary Kite codes, Z-Kite codes, MPSK-Kite codes and

Lattice-Kite codes, are considered in this paper. Numerical

results showed that the proposed codes perform well over

AWGN channels.

REFERENCES

[1] J. Byers, M. Luby, M. Mitzenmacher, and A. Rege, “A digital fountainapproaches to reliable distribution of bulk data,” in Proc. ACM SIG-

COMM’98, (Vancouver, BC, Canada), pp. 56–67, Jan. 1998.

[2] M. Luby, “LT-codes,” in Proc. 43rd Annu. IEEE Symp. Foundations of

computer Science (FOCS), (Vancouver, BC, Canada), pp. 271–280, Nov.2002.

[3] A. Shokrollahi, “Raptor codes,” IEEE Trans. Inform. Theory, vol. 52,pp. 2551–2567, Jun. 2006.

[4] D. MacKay, “Fountain codes,” IEE Proc.-Commun., vol. 152, pp. 1062–1068, Dec. 2005.

[5] R. Palanki and J. Yedidia, “Rateless codes on noisy channels,” in Proc.

2004 IEEE Int. Symp. Inform. Theory, (Chicago, IL), p. 37, June/July2004.

[6] O. Etesami and A. Shokrollahi, “Raptor codes on binary memorylesssymmetric channels,” IEEE Trans. Inform. Theory, vol. 52, pp. 2033–2051, May 2006.

[7] P. Pakzad and A. Shokrollahi, “Design principles for Raptor codes,” inProc. 2006 IEEE Inform. Theory Workshop, (Punta del Este, Uruguay),pp. 165–169, Mar. 2006.

[8] X. Ma, K. Zhang, B. Bai, and X. Zhang, “Serial concatenation ofRS codes with Kite codes: Performance analysis, iterative decodingand design.” Submitted to IEEE Trans. Inform. Theory, see alsohttp://eprintweb.org/S/article/cs/1104.4927, April 2011.

[9] G. D. Forney Jr. and M. Trott, “The dynamics of group codes:state spaces, trellis diagrams, and canonical encoders,” IEEE Trans.

Infor. Theory, vol. 39, pp. 1491 –1513, Sep. 1993.

[10] N. Sommer, M. Feder, and O. Shalvi, “Low-density lattice codes,” IEEE

Trans. Inform. Theory,, vol. 54, pp. 1561 –1585, Apr. 2008.

[11] Z.-X. Wan, Lectures on finite fields and galois rings. World ScientificPublishing Company, Oct 2003.

[12] G. D. Forney Jr., “Codes on graphs: Normal realizations,” IEEE

Trans. Inform. Theory, vol. 47, pp. 520–548, Feb. 2001.[13] X. Ma and B. Bai, “A unified decoding algorithm for linear codes based

on partitioned parity-check matrices,” in Information Theory Workshop,

2007. ITW ’07. IEEE, pp. 19 –23, Sep. 2007.[14] T. Richardson and R. Urbanke, “The capacity of low-density par-

ity check codes under message-passing decoding,” IEEE Trans. In-

form. Theory, vol. 47, pp. 599–618, Feb. 2001.[15] G. D. Forney Jr., R. Gallager, G. Lang, F. Longstaff, and S. Qureshi, “Ef-

ficient modulation for Band-limited channels,” IEEE Journal. Selected

Areas in Communications, vol. 2, pp. 632 – 647, Sep 1984.[16] X. Ma and L. Ping, “Coded modulation using superimposed binary

codes,” IEEE Trans. Inform. Theory, vol. 50, pp. 3331 – 3343, Dec.2004.

[17] J. Conway and N. Sloane, “A fast encoding method for lattice codes andquantizers,” IEEE Trans. Inform. Theory, vol. 29, pp. 820 – 824, Nov.1983.

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