Finite Frobenius Rings as a Setting for Algebraic Coding ...homepages.wmich.edu/~jwood/eprints/HFUT-Hefei.pdf · Finite Frobenius Rings as a Setting for Algebraic Coding Theory Jay

Finite Frobenius Rings as a Setting forAlgebraic Coding Theory

Jay A. Wood

Department of MathematicsWestern Michigan University

http://homepages.wmich.edu/∼jwood/

Hefei University of Technology, Hefei, AnhuiJune 30, 2011

Acknowledgments

I I am pleased to be visiting the city of Hefei, and Ithank my host, Professor Shixin Zhu, for theinvitation and for his hospitality.

JW (WMU) Finite Frobenius Rings June 30, 2011 2 / 30

Two Classical Theorems of MacWilliams

I Traditionally, linear codes are defined as linearsubspaces C ⊂ Fn

q over a finite field Fq.

I Two theorems of MacWilliams, both in her 1962Harvard doctoral dissertation, provide a foundationand an important tool for further research.

I The MacWilliams extension theorem provides thefoundation for code equivalence.

I The MacWilliams identities relate the weightenumerators of a linear code and its dual code. Theidentities are an important tool for studying linearcodes, especially self-dual codes.


Linear Codes Defined over Finite Rings

I Let R be a finite ring with 1. A linear code of lengthn defined over R is a left R-submodule C ⊂ Rn.

I There were some results on codes over rings in the1970s, but the real breakthrough came in 1994.Hammons, Kumar, Calderbank, Sloane, and Soleshowed that important duality properties of certainnon-linear binary codes could be explained by linearcodes defined over Z/4Z.

I Do the fundamental results of MacWilliams remaintrue over finite rings?


Code Equivalence

I When should two linear codes be considered thesame?

I Monomial equivalence (external)

I Linear isometries (internal)

I These notions are the same over finite fields: theMacWilliams extension theorem.


Monomial equivalence

I Work over a finite ring R .

I A permutation σ of {1, . . . , n} and invertibleelements (units) u1, . . . , un in R determine amonomial transformation T : Rn → Rn by

T (x1, . . . , xn) = (xσ(1)u1, . . . , xσ(n)un).

I Two linear codes C1,C2 ⊂ Rn are monomiallyequivalent if there exists a monomial transformationT such that C2 = T (C1).


Linear Isometries

I The Hamming weight wt(x) of a vectorx = (x1, . . . , xn) ∈ Rn is the number of nonzeroentries in x .

I A linear isomorphism f : C1 → C2 between linearcodes C1,C2 ⊂ Rn is an isometry if it preservesHamming weight: wt(f (x)) = wt(x), for all x ∈ C1.

I If T is a monomial transformation with C2 = T (C1),then the restriction of T to C1 is an isometry.

I Is the converse true? Does every linear isometrycome from a monomial transformation?


MacWilliams Extension Theorem overFinite Fields

Assume C1,C2 are linear codes in Fnq. If a linear

isomorphism f : C1 → C2 preserves Hamming weight,then f extends to a monomial transformation of Fn

q.

I MacWilliams (1961); Bogart, Goldberg, Gordon(1978)

I Ward, Wood (1996)


Characters of Finite Abelian Groups

I Let (G ,+) be a finite abelian group.

I A character π of G is a group homomorphismπ : (G ,+)→ (C×,×), where (C×,×) is themultiplicative group of nonzero complex numbers.

I Example: let G = Z/nZ be the integers modulo n.For any a ∈ Z/nZ, πa(x) = exp(2πiax/n), x ∈ G , isa character of G .

I Example: let G = Fq. For any a ∈ Fq,πa(x) = exp(2πi Tr(ax)/p), x ∈ Fq, is a characterof Fq. (Tr : Fq → Fp is the absolute trace to theprime subfield.)


Character Group

I The set G of all characters of G is itself a finiteabelian group under pointwise multiplication offunctions.

I G is called the character group.

I G ∼= G (not naturally); in particular, |G | = |G |.

IG ∼= G (naturally).

I As elements of the vector space of all functions fromG to C, the characters are linearly independent.


Two Useful Formulas

∑x∈G

π(x) =

{|G |, π = 1,

0, π 6= 1.

∑π∈G

π(x) =

{|G |, x = 0,

0, x 6= 0.


Proof of Extension Theorem (a)

I View Ci as the image of a linear map gi : Fkq → Fn

q,with g2 = f ◦ g1.

I Component functionals gi = (gi ,1, . . . , gi ,n).

I Observe, for x ∈ Fkq:

wt(gi(x)) = n −n∑

j=1

1

q

∑π∈Fq

π(gi ,j(x)).


Proof of Extension Theorem (b)

I Denote the characters of Fq by πa, a ∈ Fq.

I Weight preservation yields, for all x ∈ Fkq,

n∑j=1

∑a∈Fq

πa(g1,j(x)) =n∑

l=1

∑b∈Fq

πb(g2,l(x)).

I This is an equation of characters of Fkq.

I Use πa(x) = π1(ax) and linear independence ofcharacters to match up terms (with care).


Why Did this Proof Work?

I One key step was to replace∑

π∈Fqπ(gi ,j(x)) with∑

a∈Fqπa(gi ,j(x)).

I For a finite ring R , is R ∼= R (as one-sidedmodules)?


Finite Frobenius Rings

I Finite ring R with 1.

I The (Jacobson) radical Rad(R) of R is theintersection of all maximal left ideals of R ; Rad(R)is a two-sided ideal of R .

I The (left) socle Soc(R) of R is the ideal of Rgenerated by all the simple left ideals of R .

I R is Frobenius if R/Rad(R) ∼= Soc(R) as one-sidedmodules.


Two Useful Theorems About FiniteFrobenius Rings

I (Honold, 2001) R/Rad(R) ∼= Soc(R) as leftmodules iff R/Rad(R) ∼= Soc(R) as right modules .

I R is Frobenius iff R ∼= R as left modules iff R ∼= Ras right modules (1999).

I Corollary: R is Frobenius iff there exists a characterπ of R such that ker π contains no nonzero left(right) ideal of R . This π is a generating character.


Examples of Finite Frobenius Rings

I Finite fields Fq: π(x) = exp(2πi Tr(x)/p).

I Z/nZ: π(x) = exp(2πix/n).

I Galois rings (Galois extensions of Z/pmZ).

I Finite chain rings (all ideals form a chain).

I Products of Frobenius rings.

I Matrix rings over a Frobenius ring: Mn(R).

I Finite group rings over a Frobenius ring: R[G ].

I F2[X ,Y ]/(X 2,XY ,Y 2) is not Frobenius (Klemm,1989).


MacWilliams Extension Theorem overFinite Rings

I Theorem (1999). Let R be a finite Frobenius ring.Assume C1,C2 are linear codes in Rn. If a linearisomorphism f : C1 → C2 preserves Hammingweight, then f extends to a monomialtransformation of Rn.

I Because R ∼= R , the same proof works. (Need sometechnical details to make the matching work.)


Converse of the Extension Theorem

I Theorem (2008). If R is a finite ring and everylinear isometry between linear codes extends to amonomial transformation of Rn, then R isFrobenius.

I Earlier results in 2004–2005 by Dinh andLopez-Permouth in special cases and a generalstrategy proposed.

I A non-Frobenius ring has a left ideal of the formMm,k(Fq), with m < k . One can build acounter-example from that.


Some Notation

I We now discuss dual codes and the MacWilliamsidentities, first over finite fields.

I Finite field Fq with q elements; q a prime power.

I Dot product on Fnq (all operations in Fq):

x · y =∑

xiyi ,

for x = (x1, . . . , xn), y = (y1, . . . , yn) ∈ Fnq.

I It is a nondegenerate, symmetric bilinear form.


Definitions

I The Hamming weight wt(x) of a vectorx = (x1, . . . , xn) ∈ Fn

q is the number of nonzeroentries in x .

I A linear code over Fq of dimension k and length n isa k-dimensional vector subspace C ⊂ Fn

q.

I If C ⊂ Fnq is a linear code, then the dual code is

C⊥ = {y ∈ Fnq : x · y = 0, all x ∈ C}.


Hamming Weight Enumerators

I Given a linear code C ⊂ Fnq, the Hamming weight

enumerator of C is the two-variable polynomial(generating function):

WC (X ,Y ) =∑x∈C

X n−wt(x)Y wt(x).

I WC (X ,Y ) =∑n

i=0 AiXn−iY i , where Ai is the

number of elements of C of weight i .


“Standard Properties” of Dual Codes

1. C⊥ ⊂ Fnq.

2. C⊥ is a linear code.

3. dim C + dim C⊥ = n; or |C ||C⊥| = |Fnq|.

4. (C⊥)⊥ = C .

5. The MacWilliams identities hold:

WC⊥(X ,Y ) =1

|C |WC (X + (q − 1)Y ,X − Y ).


Ideas of Proofs

I The first four items are obvious or follow from basiclinear algebra of nondegenerate forms over fields.

I There is a proof of the MacWilliams identities dueto Gleason that is based on character theory and thePoisson summation formula.

I One key step is to identify C⊥ with thecharacter-theoretic annihilator(Fn

q : C ) := {π ∈ Fnq : π(C ) = 1}.


What Happens over Finite Rings?

I Dot product on Rn (all operations in R):

x · y =∑

xiyi ,

for x = (x1, . . . , xn), y = (y1, . . . , yn) ∈ Rn.

I Non-degenerate, but not usually symmetric when Ris non-commutative.

I Two annihilators of left submodule C ⊂ Rn:I l(C ) = {y ∈ Rn : y · x = 0, x ∈ C} (left)I r(C ) = {y ∈ Rn : x · y = 0, x ∈ C} (right)


Problems

I Always have C ⊂ l(r(C )) and D ⊂ r(l(D)).

I Equality may not hold, even when R iscommutative.

I Equality always holding is equivalent to R beingquasi-Frobenius (QF).

I QF is also equivalent to R being self-injective(injective as a module over itself).


More Problems

I Even when R is QF, one may not have |C ||r(C )|equal to |Rn|.

I Which would mean that the MacWilliams identitiesalso fail: just set X = Y = 1.

I Need R to be Frobenius, especially R ∼= R .


The MacWilliams Identities

I When R is Frobenius, one can identify (Rn : C )with r(C ), and Gleason’s proof of the MacWilliamsidentities goes through.

I The MacWilliams identities hold:

Wr(C )(X ,Y ) =1

|C |WC (X + (|R | − 1)Y ,X − Y ).


References

I These slides and other papers are available on theweb: http : //homepages.wmich.edu/ ∼ jwood

I Many references in the paper “Foundations ofLinear Codes ... ”


Thank You

I I thank again my host Professor Shixin Zhu.

I I thank you, the audience members, for your kindattention.


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Finite Frobenius Rings as a Setting for Algebraic Coding ...homepages.wmich.edu/~jwood/eprints/HFUT-Hefei.pdf · Finite Frobenius Rings as a Setting for Algebraic Coding Theory Jay