Matrix characterization of near-MDS codes over Z/sub m

Matrix Characterization of Near-MDS Codes over Z

m

and Abelian Groups

G. Viswanath and B. Sundar Rajan¹ Department of Electrical Communication Engineering

Indian Institute of Science Bangalore, 560 012, India

email: {gviswa@protocol., bsrajan@}ece.iisc.ernet.in

Abstract — In this paper we present a matrix char- acterization of AM DS codes and N M DS codes over Zm and Abelian groups.

I.AMDS andNMDS Codes overZm

A length-nlinear codeCoverZm, the ring of integers mod- ulom, is a subset ofZmⁿ closed under allZm-linear combina- tions. Cis said to be information set supporting if|C|=m^k for some integer k < nand there arekcoordinate positions such that the restriction of|C|to these kpositions isZm^k, in which caseC is referred as an [n, k, d] code overZm whered stands for the minimum Hamming distance ofC.

Definition 1 An [n, k, d] code C over Zm is said to be an Almost-MDS(AM DS)code if thed=n−k. It is said to be Near-MDS(N M DS)ifd=n−kand the minimum Hamming distance of the dual code ofC isk.

Theorem 1 An [n, k, d] linear code over Zm, where m = p^r₁¹p^r₂²p^r₃³. . . p^rs^s (wherepi1≤i≤sare prime numbers) with systematic generator matrix G = [Ik Pk,n−k] (after suitable column permutations) is anAM DS code if and only if every (g, g+ 1)submatrix has(g, g)submatrices such that

• there exists at least one(g, g)submatrix whose determi- nant is a unit inZm or

• if the determinants of all the(g, g)submatrices are zero divisors then the greatest common divisor of these de- terminants is an unit inZm.

II.AMDS andNMDS Codes over Abelian Groups

Definition 2 An [n, k] systematic group code C of length n and dimension k over an Abelian group G is a subgroup of Gⁿ with order |G|^k consisting of n-tuples, (x1, x2,· · ·, xk, y1,· · ·, yn−k) with yi = φi(x1, x2, . . . , xk) whereφiare(n−k)homomorphisms fromG^kintoG.

Definition 3 Consider a finite Abelian group G with cardi- nalitym. An[n, k=logm(|C|), d]codeCoverGis an Almost- MDSAM DS code ifd=n−k. An AMDS codeCis said to be Near-MDS (N M DS) if its dual code [2] is alsoAM DS.

Theorem 2 A[k+s, k]group code overG, defined by the ho- momorphisms{φ1, φ2, . . . , φs}isAM DS if and only if every (g, g+ 1)submatrix of the associated matrix of the form

Ψg,g+1= 2 66 64

ψi₁j₁ ψi₁j₂ . . . ψi₁j_g+1

ψi₂j₁ ψi₂j₂ . . . ψi₂j_g+1

... ... . . . ... ψi_gj₁ ψi_gj₂ . . . ψi_gj_g+1

3 77

75 (1)

1This work was partly supported by the DRDO-IISc Program on Advanced Research in Mathematical Engineering through a grant to B.S.Rajan

for1≤ik≤g,1≤jk≤(g+ 1),g= 1,2, . . . , min{s, k}, has

• at least one(g, g)submatrix which represents an auto- morphism ofG^g or

• if every(g, g)submatrix represents an endomorphism of G^g then the intersection of the kernels of the endomor- phisms is only the identity element ofG^g.

We specialize to the case where the groupGis a cyclic group Cm, i.e., a cyclic group of orderm.

Definition 4 An homomorphism φ: Cm^k → Cm is called a distance non decreasing homomorphism (DN DH) if either Kφ={e}ordmin(Kφ) = 1, whereeis the identity element of CM^k,Kφdenotes the kernel ofφanddminstands for minimum Hamming distance.

Lemma 3 A[k+ 1, k] group code isAM DS if and only if the defining homomorphism is anDN DH.

Definition 5 Let {φ}^i=si=1 denoted as Φs be a set of homo- morphisms from Cm^k → Cm denoted as Φ_(s). Let Kφ₁φ₂...φ_s

denoteKφ₁∩Kφ₂∩. . . Kφ_swhereKφ_iis the kernel ofφi. Φs

is said to be a distance non decreasing set of homomorphisms, (DN DSH), if the following conditions are satisfied:

• the homomorphisms do not constitute a set of DISH [2]

• for all1≤r≤s dmin(Kφ_i1φ_i2...φ_ir)≥ror

• Kφ_i1φ_i2...φ_ir ={e}.

Theorem 4 A[k+s, k]group code isAM DSif and only if the defining homomorphismsΦsconstitute a set ofDN DSH.

Lemma 5 Over Cm, where m = p^d₁¹p^d₂²p^d₃³. . . p^dr^r, where pi’s are distinct primes, [k+s, s] AM DS group codes, for all s, k ≥ 2, do not exist if k ≥ r+ 2(p−1), where p = min{p1, p2, . . . , pr}.

Lemma 6 Over Cm, where m= p^d₁¹p^d₂²p^d₃³. . . p^dr^r, with all primespi distinct, the dual of[k+s, s]AM DS group codes, for alls, k ≥ 2do not exist if s ≥ r+ 2(p−1), where p = min{p1, p2, . . . , pr}.

References

[1] S. M. Dodunekov and I. N. Landgev, ’On Near-MDS Codes’, Technical Report, No:LiTH-ISY-R-1563, Department of Elec- trical Engineering, Linkoping University, February, 1994.

[2] A. A. Zain and B. Sundar Rajan, ’Algebraic Characterization of MDS Group Codes over Cyclic Groups’,IEEE Trans. Infor- mation Theory, Vol.41, No.6, Nov., 1997, pp.2052-2056.

ISIT 2004, Chicago, USA, June 27 – July 2, 2004

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