Matrix characterization of linear codes with arbitrary Hamming weight hierarchy

The support of an [n, k] linear code C over a finite field $F_{q}$ 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 $d_{r}$(C), 1 \leq r \leq k, of C is defined as the minimum of the cardinalities of the supports of all [n, r] subcodes of C. The sequence $(d_{1}(C), d_{2}(C),..., d_{k}(C))$ is called the Hamming weight hierarchy (HWH) of C. The HWH, $d_{r}$(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) X (n - k) identity matrix and P is a (n - k) X k matrix, is MDS iff every square submatrix of P is nonsingular. In this paper we extend this characterization to linear codes with arbitrary HWH. Using this result, we characterize Near-MDS codes, Near-Near-MDS $(N^{2}-MDS)$ codes and $A^{\mu}-MDS$ codes. The MDS-rank of C is the smallest integer n such that $d_{n+1} = n - k + n + 1$ and the defect vector of C with MDS-rank n is defined as the ordered set ${\mu1(C), \mu2(C), \mu3(C),..., \mu_{n}(C), \mu_{n+1}(C)}$, where $\mu_{i}$(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 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.