The Weights in MDS Codes

Abstract

The weights in MDS codes of length n and dimension k over the finite field GF(q) are studied. Up to some explicit exceptional cases, the MDS codes with parameters given by the MDS conjecture are shown to contain all k weights in the range n-k+1 to n. The proof uses the covering radius of the dual codeComment: 5 pages, submitted to IEEE Trans. IT. This version 2 is the revised version after the refereeing process. Accepted for publicatio

New construction of quantum error-avoiding codes via group representation of quantum stabilizer codes

“…To connect and compare quantum error-correcting codes with classic error-correcting codes, we thenceforth denote a classic code that encodes k binary bits to n binary bits as an [n, k] code. Furthermore, we denote a quantum code that encodes k qubits to n qubits as an [26,27]. If a code has minimum distance d, it is capable of detecting any d − 1 and correcting any…”

Pure Asymmetric Quantum MDS Codes From CSS Construction: A Complete Characterization

“…Let C be an [n, k, n − k + 1] q -MDS code. Ezerman et al [5] showed that C has a codeword u with wt(u) = n, except when either C is the dual of the binary repetition code of odd length n ≥ 3, or C is a simplex code with parameters [q + 1, 2, q] q . Hence, the following corollary can be derived.…”

Optimal frequency hopping sequence sets from MDS codes

“…Remark 1: When n = q+1, then the MDS codes in Proposition 1 are those codes given by Theorem 8 of [7]. When n < q + 1, they can be regarded as the codes are punctured from those codes given by Theorem 8 of [7].…”

“…In [7], [14], [24], some properties of MDS codes of length q +1 over the finite field F q were considered, where q is a prime power. In this paper, we consider those punctured MDS codes.…”

“…In Section 2, we introduce some definitions and several known bounds on the parameters of FH sequence sets. In Section 3, we give the punctured codes from the MDS codes given by [7], [14], [24], and also present the corresponding FH sequence sets from those punctured MDS codes, the parameters of which are optimal with respect to the lower bound from Singleton bound. Finally, some concluding remarks are given in Section 4.…”