Recently, Yaakobi et al. introduced codes for b-symbol read channels, where the read operation is performed as a consecutive sequence of b > 2 symbols. In this paper, we establish a Singleton-type bound on b-symbol codes. Codes meeting the Singleton-type bound are called maximum distance separable (MDS) codes, and they are optimal in the sense they attain the maximal minimum bdistance. Based on projective geometry and constacyclic codes, we construct new families of linear MDS b-symbol codes over finite fields. And in some sense, we completely determine the existence of linear MDS b-symbol codes over finite fields for certain parameters.
Motivated by the application of high-density data storage technologies, symbol-pair codes are proposed to protect against pair-errors in symbol-pair channels, whose outputs are overlapping pairs of symbols. The research of symbol-pair codes with the largest minimum pair-distance is interesting since such codes have the best possible error-correcting capability. A symbol-pair code attaining the maximal minimum pair-distance is called a maximum distance separable (MDS) symbol-pair code. In this paper, we focus on constructing linear MDS symbol-pair codes over the finite field F q . We show that a linear MDS symbol-pair code over F q with pair-distance 5 exists if and only if the length n ranges from 5 to q 2 + q + 1. As for codes with pair-distance 6, length ranging from 6 to q 2 +1, we construct linear MDS symbol-pair codes by using a configuration called ovoid in projective geometry. With the help of elliptic curves, we present a construction of linear MDS symbol-pair codes for any pair-distance d + 2 with length n satisfying 7 ≤ d + 2 ≤ n ≤ q + ⌊2 √ q⌋ + δ(q) − 3, where δ(q) = 0 or 1.
In this paper we study a certain generalization of combinatorial designs related to almost difference sets, namely the t-adesign, which was coined by Cunsheng Ding in 2015. It is clear that 2-adesigns are a kind of partially balanced incomplete block design which naturally arise in many combinatorial and statistical problems. We discuss some of their basic properties and give several constructions of 2-adesigns (some of which correspond to new almost difference sets, and others of which correspond to new almost difference families), as well as two constructions of 3-adesigns. We also discuss some basic properties of their incidence matrices and codes.
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