linear codes over finite rings and moduleshomepages.wmich.edu/~jwood/eprints/wuhan-06.pdfep for...

Linear Codes over Finite Rings andModules

Jay A. Wood

Department of MathematicsWestern Michigan University

http://homepages.wmich.edu/∼jwood

Central China Normal UniversityWuhan, HubeiMay 7, 2018

MacWilliams extension theorem

6. MacWilliams extension theorem

I Extension property (EP)

I EP for Hamming weight over Frobenius bimodulesvia linear independence of characters

I EP for Hamming weight over Frobenius rings

I Generalization for module alphabets

JW Codes May 7, 2018 2 / 31


Notation

I Let R be a finite associative ring with 1.

I Let A be a finite unital left R-module: thealphabet.

I Let w : A→ Q be a weight: w(0) = 0. Extend toAn by

w(a1, . . . , an) =n∑

i=1

w(ai).

JW Codes May 7, 2018 3 / 31


Convention

I I will usually write homomorphisms of left moduleson the right side: inputs on the left.

I A homomorphism f : A→ A satisfies

(a1 + a2)f = a1f + a2f ,

(ra)f = r(af ),

for r ∈ R and a, a1, a2 ∈ A.

JW Codes May 7, 2018 4 / 31


Symmetry groups

I Define the symmetry groups of w :

Glt = {u ∈ U(R) : w(ua) = w(a), a ∈ A},Grt = {φ ∈ GLR(A) : w(aφ) = w(a), a ∈ A}.

I U(R) is the group of units of R , and GLR(A) is thegroup of invertible R-linear homomorphisms A→ A.

JW Codes May 7, 2018 5 / 31


Monomial transformations

I For a subgroup G ⊆ GLR(A), a G -monomialtransformation of An is an invertible R-linearhomomorphism T : An → An of the form

(a1, a2, . . . , an)T = (aσ(1)φ1, aσ(2)φ2, . . . , aσ(n)φn),

for (a1, a2, . . . , an) ∈ An.

I Here, σ is a permutation of {1, 2, . . . , n} andφi ∈ G for i = 1, 2, . . . , n.

JW Codes May 7, 2018 6 / 31


Isometries

I Let C1,C2 ⊆ An be two linear codes. An R-linearisomorphism f : C1 → C2 is a linear isometry withrespect to w if w(xf ) = w(x) for all x ∈ C1.

I Every Grt-monomial transformation is an isometryfrom An to itself.

JW Codes May 7, 2018 7 / 31


Extension property (EP)

I Given ring R , alphabet A, and weight w on A.

I The alphabet A has the extension property (EP)with respect to w if the following holds: For any leftlinear codes C1,C2 ⊆ An, if f : C1 → C2 is a linearisometry, then f extends to a Grt-monomialtransformation An → An.

I That is, there exists a Grt-monomial transformationT : An → An such that xT = xf for all x ∈ C1.

JW Codes May 7, 2018 8 / 31


Slightly different point of view

I Linear codes are often presented by generatormatrices. A generator matrix serves as a linearencoder from an information space to a messagespace.

I If f : C1 → C2 is a linear isometry, then C1 and C2

are isomorphic as R-modules. Let M be a leftR-module isomorphic to C1 and C2. Call M theinformation module.

I Then C1 and C2 are the images of R-linearhomomorphisms Λ : M → An and N : M → An,respectively. We have N = Λf : inputs on left!

JW Codes May 7, 2018 9 / 31


Coordinate functionals

I C1 was given by Λ : M → An. Write the individualcomponents as Λ = (λ1, . . . , λn), withλi ∈ HomR(M ,A). Call the λi coordinatefunctionals.

I Similarly, N = (ν1, . . . , νn), νi ∈ HomR(M ,A).

I The isometry f extends to a Grt-monomialtransformation if there exists a permutation σ andφi ∈ Grt such that νi = λσ(i)φi for all i = 1, . . . , n.

JW Codes May 7, 2018 10 / 31


Case of R

I Our first result will show that, for any finite ring R ,A = R has EP with respect to the Hamming weight.

I It follows that A = R itself has EP with respect tothe Hamming weight when R is Frobenius.

I The Frobenius ring case came first (1999).

I The more general A = R case is due to Greferath,Nechaev, and Wisbauer (2004).

JW Codes May 7, 2018 11 / 31


Techniques

I For any alphabet A, the summation formulas forcharacters imply that the Hamming weight wtsatisfies

wt(a) = 1− 1

|A|∑π∈A

π(a), a ∈ A.

I Characters (in multiplicative form) are linearlyindependent functions on A over C (Lecture 9).

I Recursive argument using maximal elements in afinite poset.

JW Codes May 7, 2018 12 / 31


Symmetry groups for the Hamming weight

I Consider the Hamming weight wt on A = R , whichis an (R ,R)-bimodule.

I Both symmetry groups Glt and Grt equal U(R).

JW Codes May 7, 2018 13 / 31


Posets

I Given a set S , a (non-strict) partial order � on Sis reflexive, antisymmetric, and transitive. The pair(S ,�) is a partially ordered set or poset.

I Example. Let X be a nonempty set. ThenS = P(X ), the set of all subsets of X , is a posetunder set inclusion, i.e., U � V when U ⊆ V .

JW Codes May 7, 2018 14 / 31


Poset of cyclic submodules

I Example. Let B be a finite right R-module. ThenS = {bR : b ∈ B} is the poset of all cyclic rightR-submodules of B under set inclusion.

I Fact: For finite rings R , b1R = b2R if and only ifb1 = b2u, where u ∈ U(R).

I This fact uses of work of Bass on rings of stablerange one.

JW Codes May 7, 2018 15 / 31


Proof of EP for A = R (i)

I Same set up as before: ring R , alphabet A = R ,with Hamming weight.

I C1,C2 ⊆ Rn, with f : C1 → C2 linear isometry.

I C1 is image of Λ : M → Rn; C2 is image ofN : M → Rn. N = Λf .

I Isometry: wt(xΛ) = wt(xN), for all x ∈ M .

JW Codes May 7, 2018 16 / 31


Proof (ii)

I Remember, A = R .

I A has a left generating character: ρ : A→ C,ρ(π) = π(1) for π ∈ R . (Evaluate at 1 ∈ R .)

I Every character of A (element of A) has the form rρfor some unique r ∈ R .

I Recall: (rρ)(a) = ρ(ar).

JW Codes May 7, 2018 17 / 31


Proof (iii)

I Hamming weight as character sum:

n∑i=1

∑r∈R

rρ(xλi) =n∑

j=1

∑s∈R

sρ(xνj), x ∈ M .

I That is,

n∑i=1

∑r∈R

ρ(xλi r) =n∑

j=1

∑s∈R

ρ(xνjs), x ∈ M .

I This is an equation of characters on M .

JW Codes May 7, 2018 18 / 31


Proof (iv)

I Let B = HomR(M ,A), a right R-module. PosetS = {λR : λ ∈ HomR(M ,A)} under ⊆.

I Among the λiR , νjR , choose one that is maximal for⊆. Say, ν1R .

I Let j = 1 and s = 1 on the right side of thecharacter equation:

n∑i=1

∑r∈R

ρ(xλi r) =n∑

j=1

∑s∈R


JW Codes May 7, 2018 19 / 31


Proof (v)

I Let j = 1 and s = 1 on the right side of thecharacter equation:

n∑i=1

∑r∈R

ρ(xλi r) =n∑

j=1

∑s∈R


I By linear independence of characters, there exists i1and r ∈ R so that ρ(xλi1r) = ρ(xν1) for all x ∈ M .

I Thus ρ(x(ν1 − λi1r)) = 1 for all x ∈ M .

I That is, M(ν1 − λi1r) ⊆ ker ρ.

JW Codes May 7, 2018 20 / 31


Proof (vi)

I We had M(ν1 − λi1r) ⊆ ker ρ.

I M(ν1 − λi1r) is a left R-module.

I Because ρ a generating character, ν1 = λi1r .

I Thus, ν1 ∈ λi1R and ν1R ⊆ λi1R .

I By maximality of ν1R , ν1R = λi1R .

I Thus, ν1 = λi1u1, for some u1 ∈ U(R).

JW Codes May 7, 2018 21 / 31


Proof (vii)

I Recall,

n∑i=1

∑r∈R

ρ(xλi r) =n∑

j=1

∑s∈R


I Then inner sums agree:∑r∈R ρ(xλi1r) =

∑s∈R ρ(xν1s), x ∈ M .

I Set σ(1) = i1. Subtract inner sums to reduce thesize of the outer sums by 1. Proceed by induction.

JW Codes May 7, 2018 22 / 31


Slightly more general result

I The exact same proof applies to a Frobeniusbimodule, a bimodule A over R such that A isisomorphic to R as left R-modules and as rightR-modules (but not necessarily isomorphic asbimodules).

JW Codes May 7, 2018 23 / 31


Generalize to module alphabets

I For ring R , alphabet A, and Hamming weight wt,EP holds if A: (1) is pseudo-injective and (2) has a

cyclic socle (embeds into R).

I Pseudo-injective means injective with respect tosubmodules. That is, if B is a submodule of A andh : B → A is any injective module homomorphism,then h extends to injective h : A→ A.

JW Codes May 7, 2018 24 / 31


EP for linear codes of length 1

I Dinh, Lopez-Permouth: EP for linear codes over Aof length 1 is equivalent to A being pseudo-injective.

I A linear code of length 1 is a submodule C of A.

I Any injection C ↪→ A preserves Hamming weight.

I EP reduces to the algebraic question of whether aninjection of a submodule always extends to all of A:pseudo-injectivity.

JW Codes May 7, 2018 25 / 31


EP for Hamming weight over modulealphabets

TheoremSuppose a left R-module A is pseudo-injective and has acyclic socle. Then A has EP with respect to theHamming weight.

I Because Soc(A) is cyclic, A embeds into R .

JW Codes May 7, 2018 26 / 31


Proof (i)

I Suppose C1,C2 ⊂ An are R-linear codes withisomorphism f : C1 → C2 that preserves theHamming weight on An.

I Via A ↪→ R , view C1,C2 ⊆ Rn.

I The Hamming weight of x ∈ An ⊆ Rn is the same,whether x is considered as an element of An or asan element of Rn.

I So, C1,C2 ⊆ Rn, with f : C1 → C2 preserving theHamming weight from Rn.

JW Codes May 7, 2018 27 / 31


Proof (ii)

I R has EP with respect to Hamming weight.

I So f : C1 → C2 extends to a monomialtransformation F of Rn.

I Write (x1, . . . , xn)F = (xσ(1)u1, . . . , xσ(n)un), for

(x1, . . . , xn) ∈ Rn, where σ is a permutation of

{1, 2, . . . , n} and ui ∈ U(R) = Aut(RR).

I Does ui map A back to A? No way to tell.

I Write F = PD, where P is the permutation partand D is the diagonal part.

JW Codes May 7, 2018 28 / 31


Proof (iii)

I Set C3 = C1P ⊆ An ⊆ Rn.

I Then D maps C3 → C2 and preserves Hammingweight.

I Look at individual components of D.

I For i = 1, . . . , n, project C3,C2 to the ith entry,

C(i)3 ,C

(i)2 ⊆ A ⊆ R .

I Define D(i) by xD(i) = xui , x ∈ R . Then D(i) maps

C(i)3 → C

(i)2 and preserves the Hamming weight.

JW Codes May 7, 2018 29 / 31


Proof (iv)

I Recall, C(i)3 ,C

(i)2 ⊆ A and D(i) maps C

(i)3 → C

(i)2

injectively. In particular, D(i) : C(i)3 ↪→ A.

I Because A is pseudo-injective, D(i) extends to anautomorphism τi ∈ Aut(RA).

I Then F ′, defined by(x1, . . . , xn)F ′ = (xσ(1)τ1, . . . , xσ(n)τn), is a monomialtransformation of An extending f .

JW Codes May 7, 2018 30 / 31


What is coming next?

I Converses!

I If a ring alphabet R has EP with respect to theHamming weight, then R is Frobenius.

I If a module alphabet A has EP with respect to theHamming weight, then A is pseudo-injective and hasa cyclic socle.

JW Codes May 7, 2018 31 / 31

linear codes over finite rings and moduleshomepages.wmich.edu/~jwood/eprints/wuhan-06.pdfep for...

Documents