More on the Stopping and Minimum Distances of Array Codes

For q an odd prime and 1≤ m ≤ q, two specific binary qm × q
²
parity-check matrices denoted by H
_P
(m, q) and H
_I
(m, q) are considered. The corresponding binary codes, C
_P
(m, q) and C
_I
(m, q), respectively, are called proper and improper array codes with parameters m and q. Given a parity-check matrix H representing a binary code C, let s(H) denote the stopping distance of H and d(C) be the minimum Hamming distance of C. It is known that that s(H
_I
(m, q)) = s(H
_P
(m, q)) = d(C
_I
(m, q)) = d(C
_P
(m, q)) for m ≤ 3. In this paper, we show that these equalities do not hold for all values of m and q. In particular, although s(H
_P
(4, 7)) = d(C
_P
(4, 7)) = 8 we have s(H
_I
(4, 7)) = 9 and d(C
_I
(4, 7)) = 10. It is also shown that s(H
_P
(5,1))dC
_P
(5, 11)) = 10 while s(H
_I
(5,11)) = 11 and d(C
_I
(5, 11)) = 12. This suggests that in many cases the improper array codes would perform better than the proper array codes over the AWGN and binary erasure channels. Performance results are given which confirm this claim. The combinatorial structure of the eight-element stopping sets for H(m ≥ 4,q >; 5) is also determined.