Checkable Codes form Group Algebras toGroup Rings

Noha Abdelghany

Department of Mathematics, Faculty of Science, Cairo University.

Lens, NCRA IV

June 11, 2015

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Table of Contents

History

Group-Ring Codes

Code-Checkable Group Rings

References

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History

I (MacWilliams, 1969)"Codes and ideals in group algebras".

I (Hurley, 2007)"Module codes over group rings".

I (Hurley, 2009)"Codes from zero-divisors and units in group rings".

I (Jitman, 2010)"Checkable Codes from group rings".

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Codes Over Finite Fields

Let Fnq denote the vector space of all n-tuples over the finite

field Fq.I An (n,M) code C over Fq is a subset of Fn

q of size M.I If C is a k -dimensional vector subspace of Fn

q, then C willbe called [n,k] linear code over Fq. This linear code C hasqk codewords.

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Generator and Parity Check Matrix of a Linear Code

I A generator matrix for an [n, k ] linear code C is any k × nmatrix G whose rows form a basis for the code C. C iswritten as:

C = {xG : x ∈ Fkq}

I A parity check matrix H for an [n, k ] linear code C is an(n − k)× n matrix defined by:

x ∈ C ⇔ HxT = 0

I Note that GHT = 0. Thus G, H are in a sensezero-divisors.

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Cyclic Codes

The class of cyclic codes is one of the most important classesof codes. In fact almost all codes used for practical issues, likeBCH and Reed-Solomon codes, are cyclic codes. This is due tothe existence of fast encoding and decoding algorithms.

DefinitionA linear code C is a cyclic code if C satisfies:

(c1, c2, . . . , cn−1, cn) ∈ C ⇒ (cn, c1, . . . , cn−1) ∈ C,

For every c ∈ Fnq.

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Table of Contents

History

Group-Ring Codes

Code-Checkable Group Rings

References

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Group Rings

DefinitionGiven a group G and a ring R the group ring RG is the ringconsisting of the set of all formal finite sums

∑g∈G αgg, where

αg ∈ R.

For u =∑

g∈G αgg, v =∑

g∈G βgg ∈ RG and α ∈ R, define:I u + v =

∑g∈G(αg + βg)g,

I uv = (∑

g∈G αgg)(∑

h∈G βhh) =∑

g∈G(∑

h∈G αhβh−1g)g,

I αu =∑

g∈G(αug)g.

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Basic Properties of Group Rings

I The group ring RG is a ring.I The group ring RG is a left R-module.I When R is a field, the group ring RG is an algebra over R

and it is called group algebra.

TheoremFor a fixed listing of elements of a finite groupG = {g1,g2, . . . ,gn} there is a one-to-one correspondencebetween RG and a subring of the matrix ring Mn(R), given by:

w =n∑

i=1

αigi →W =

αg−1

1 g1αg−1

1 g2. . . αg−1

1 gn...

.... . .

...αg−1

n g1αg−1

n g2. . . αg−1

n gn

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Group-Ring Codes

Let RG be a group ring, W a submodule of RG and u ∈ RG.

DefinitionI A right group ring encoding is a mapping f : W → RG,

such that f (x) = xu.I The group-ring code C generated by u relative to W is

the image of the group ring encoding: C = {xu : x ∈W}.

If u is a zero-divisor (resp. unit), C is called zero-divisor (resp.unit-derived) code.

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Table of Contents

History

Group-Ring Codes

Code-Checkable Group Rings

References

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Checkable Codes

I Suppose that u is a zero-divisor in RG and let v be anon-zero element such that uv = 0.

I Let C = {xu : x ∈W} = Wu be a code generated by urelative to W . Then

y ∈ C ⇒ yv = 0.

I If the zero-divisor code C satisfies: y ∈ C ⇔ yv = 0. ThenC is called checkable code. In other words,

C = {y ∈ RG : yv = 0}.

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Code-Checkable Group Rings

C is said to be checkable if C = {y ∈ RG : yv = 0} for somev ∈ RG.

Definition (Jitman, 2010)A group ring RG is said to be code-checkable if every ideal inRG is a checkable code.

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Characterization of Code-Checkable Group Algebras

Let G be a finite abelian group and F be a finite field ofcharacteristic p.

Proposition (Jitman, 2010)The group algebra FG is code-checkable if and only if it is aPIR.

Theorem (Fisher and Sehgal, 1976)The group algebra FG is a PIR if and only if a Sylowp-subgroup of G is cyclic.

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Characterization of Code-Checkable Group Algebras

Let G be a finite abelian group and F be a finite field ofcharacteristic p.

Proposition (Jitman, 2010)The group algebra FG is code-checkable if and only if it is aPIR.

Theorem (Fisher and Sehgal, 1976)The group algebra FG is a PIR if and only if a Sylowp-subgroup of G is cyclic.

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Characterization of Code-Checkable Group Algebras

Theorem (Jitman, 2010)Let G be a finite abelian group and F be a finite field ofcharacteristic p. Then the group algebra FG is code-checkableif and only if a Sylow p-subgroup of G is cyclic.

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DefinitionLet π be a finite set of primes. A finite group G is calledπ′-by-cyclic π, if there is a normal subgroup H C G such that:

I |H| is coprime with each prime in π.I The quotient group G/H is cyclic and a π-group.

ExampleLet π = {2}. Since A3 C S3, |A3| = 3 and |S3/A3| = 2. Then S3is π′-by-cyclic π.

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Characterization of Code-Checkable Group Rings

LemmaLet R be a finite commutative ring and G a finite group. ThenRG is code-checkable if and only if RG is a principal idealgroup ring.

Theorem (Dorsey, 2006)Let R be a finite semisimple ring and G a finite group. Then RGis PIR if and only if G is π′-by-cyclic π, where π is the set ofnoninvertible primes in R.

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Characterization of Code-Checkable Group Rings

LemmaLet R be a finite commutative ring and G a finite group. ThenRG is code-checkable if and only if RG is a principal idealgroup ring.

Theorem (Dorsey, 2006)Let R be a finite semisimple ring and G a finite group. Then RGis PIR if and only if G is π′-by-cyclic π, where π is the set ofnoninvertible primes in R.

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Characterization of Code-Checkable Group Rings

TheoremLet G be a finite group, R a finite commutative semisimple ringand π the set of noninvertible primes in R. Then the group ringRG is code-checkable if and only if G is π′-by-cyclic π.

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Table of Contents

History

Group-Ring Codes

Code-Checkable Group Rings

References

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References I

[1] T. J. Dorsey, Morphic and Principal-Ideal Group Rings, Journal of Algebra, vol. 318,pp. 393-411, (2007).

[2] J. L. Fisher and S. K. Sehgal, Principal Ideal Group Rings, Comm. Algebra, vol. 4,pp. 319-325, (1976).

[3] W. C. Huffman and V. Pless, Fundamental of Error-Correcting Codes, CambridgeUniversity Press, New York, (2003).

[4] P. Hurley and T. Hurley, Module codes in group rings, Proc. IEEE Int. Symp. onInformation Theory, (2007).

[5] P. Hurley and T. Hurley, Codes from zero-divisors and units in group rings, Int. J.Information and Coding Theory, pp. 57-87, (2009).

[6] S. Jitman, S. Ling, H. Liu and X. Xie, Checkable Codes from Group Rings, CoRR(2011).

[7] T. Y. Lam, A First course in noncommutative rings, second ed., Graduate Texts inMathematics, vol. 131, Springer-Verlag, New York, 2001.

[8] F. J. MacWilliams, Codes and ideals in group algebras Combinatorial Mathematicsand its Applications, pp. 312-328, (1969).

[9] S. C. Misra, S. Misra and I. Woungang, Selected topics in information and codingtheory, World Scientific Publishing Co. Pte. Ltd., vol. 7, 2010.

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Thank You for Your Time.

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