Matrix characterization of linear codes with arbitrary Hamming weight hierarchy

The support of an [n, k] linear code C over a finite field Fq is the set of all coordinate positions such that at least one codeword has a nonzero entry in each of these coordinate position. The rth generalized Hamming weight dr (C), 1 r k, of C is defined as the minimum of the cardinalities of the supports of all [n, r] subcodes of C. The sequence (d1(C), d2(C), . . . , dk(C)) is called the Hamming weight hierarchy (HWH) of C. The HWH, dr (C) = n− k + r; r = 1, 2, . . . , k, characterizes maximum distance separable (MDS) codes. Therefore the matrix characterization of MDS codes is also the characterization of codes with the HWH dr (C) = n− k + r; r = 1, 2, . . . , k. A linear code C with systematic check matrix [I |P ], where I is the (n− k)× (n− k) identity matrix andP is a (n− k)× kmatrix, is MDS iff every square submatrix ofP is nonsingular. In this paper we extend this characterization to linear codes with arbitrary HWH. Using this result, we characterize Near-MDS codes, Near-NearMDS (N2-MDS) codes andAμ-MDS codes. The MDS-rank ofC is the smallest integer η such thatdη+1 = n− k + η + 1 and the defect vector ofC with MDS-rankη is defined as the ordered set {μ1(C), μ2(C), μ3(C), . . . , μη(C), μη+1(C)}, where μi(C) = n− k + i − di(C). We call C a dually defective code if the defect vector of the code and its dual are the same. We This work was partly supported by the DRDO-IISc Program on Advanced Research in Mathematical Engineering as well as the Council of Scientific and Industrial Research (CSIR), India, through research grant (22(0365)/04/EMR-II) to B.S. Rajan. Part of this work was presented in IEEE International Symposium on Information Theory held in Washington, DC, USA, June 24–29, 2001, p. 61. ∗ Corresponding author. E-mail addresses: [email protected] (G. Viswanath), [email protected] (B. Sundar Rajan). 0024-3795/$ see front matter ( 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2005.07.008 G. Viswanath, B. Sundar Rajan / Linear Algebra and its Applications 412 (2006) 396–407 397 also discuss matrix characterization of dually defective codes. Further, the codes meeting the generalized Greismer bound are characterized in terms of their generator matrix. The HWH of dually defective codes meeting the generalized Greismer bound are also reported. © 2005 Elsevier Inc. All rights reserved. AMS classification: 94B05