A note on the constructions of MDS self-dual codes

Explicit Constructions of MDS Self-Dual Codes

“…This code is an almost MDS code. We also find almost MDS self-dual codes over F 16 with parameters [20,10,10], [22,11,11], [24,12,12]. Proof.…”

“…Due to Lemma 6 and Lemma 7, new classes of selfdual codes with prescribed minimum distance are constructed. Additionally, we construct MDS self-dual codes with new parameters [24,12,13] 16,15] 16 .…”

A Unified Approach to Construct MDS Self-Dual Codes via Reed-Solomon Codes

“…We present simple proofs with the approach illustrated in Section II. For more complete list of known MDS self-dual codes we refer to the table in [3] and [15].…”

New MDS self-dual codes over finite fields of odd characteristic

“…. , a n−1 ), the extended GRS code of length n associated with − → v and − → a is defined as follows: We present another two useful results, which have been shown in [23].…”

“…All the known results on the systematic constructions of MDS self-dual codes are depicted in Table 1. Table 1: Known systematic construction on MDS self-dual codes of length n (η is the quadratic character of F q ) q n even Reference q even n ≤ q [7] q odd n = q + 1 [7] q odd (n − 1)|(q − 1), η(1 − n) = 1 [23] q odd (n − 2)|(q − 1), η(2 − n) = 1 [23] q = r s ≡ 3 (mod 4) n − 1 = p m | (q − 1), prime p ≡ 3 (mod 4) and m odd [8] q = r s , r ≡ 1 (mod 4), s odd n − 1 = p m | (q − 1), m odd and prime p ≡ 1 (mod 4) [8] q = r s , r odd, s ≥ 2 n = lr, l even and 2l|(r − 1) [23] q = r s , r odd, s ≥ 2 n = lr, l even , (l − 1)|(r − 1) and η(1 − l) = 1 [23] q = r s , r odd, s ≥ 2 n = lr + 1, l odd , l|(r − 1) and η(l) = 1 [23] q = r s , r odd, s ≥ 2 n = lr + 1, l odd , (l − 1)|(r − 1) and η(l − 1) = η(−1) = 1…”