Maximum Distance Separable Codes for Symbol-Pair Read Channels

We study (symbol-pair) codes for symbol-pair read channels introduced recently by Cassuto and Blaum (2010). A Singleton-type bound on symbol-pair codes is established and infinite families of optimal symbol-pair codes are constructed. These codes are maximum distance separable (MDS) in the sense that they meet the Singleton-type bound. In contrast to classical codes, where all known q-ary MDS codes have length O(q), we show that q-ary MDS symbol-pair codes can have length \Omega(q^2). In addition, we completely determine the existence of MDS symbol-pair codes for certain parameters

Maximum Weight Spectrum Codes

In the recent work \cite{shi18}, a combinatorial problem concerning linear codes over a finite field \F_q was introduced. In that work the authors studied the weight set of an $[n,k]_q$ linear code, that is the set of non-zero distinct Hamming weights, showing that its cardinality is upper bounded by $\frac{q^k-1}{q-1}$. They showed that this bound was sharp in the case $q=2$, and in the case $k=2$. They conjectured that the bound is sharp for every prime power $q$ and every positive integer $k$. In this work quickly establish the truth of this conjecture. We provide two proofs, each employing different construction techniques. The first relies on the geometric view of linear codes as systems of projective points. The second approach is purely algebraic. We establish some lower bounds on the length of codes that satisfy the conjecture, and the length of the new codes constructed here are discussed.Comment: 19 page

MRD codes with maximum idealisers

Left and right idealizers are important invariants of linear rank-distance codes. In the case of maximum rank-distance (MRD for short) codes in GF(q)^(n×n) the idealizers have been proved to be isomorphic to finite fields of size at most q^n. Up to now, the only known MRD codes with maximum left and right idealizers are generalized Gabidulin codes, which were first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and Gabidulin in 2005. In this paper we classify MRD codes in GF(q)^(n×n) for n9

On optimal quantum codes

We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters [[n,n-2d+2,d]]_q exist for all 3 <= n <= q and 1 <= d <= n/2+1. We also present quantum MDS codes with parameters [[q^2,q^2-2d+2,d]]_q for 1 <= d <= q which additionally give rise to shortened codes [[q^2-s,q^2-2d+2-s,d]]_q for some s.Comment: Accepted for publication in the International Journal of Quantum Informatio

Long Nonbinary Codes Exceeding the Gilbert - Varshamov Bound for any Fixed Distance

Let A(q,n,d) denote the maximum size of a q-ary code of length n and distance d. We study the minimum asymptotic redundancy \rho(q,n,d)=n-log_q A(q,n,d) as n grows while q and d are fixed. For any d and q<=d-1, long algebraic codes are designed that improve on the BCH codes and have the lowest asymptotic redundancy \rho(q,n,d) <= ((d-3)+1/(d-2)) log_q n known to date. Prior to this work, codes of fixed distance that asymptotically surpass BCH codes and the Gilbert-Varshamov bound were designed only for distances 4,5 and 6.Comment: Submitted to IEEE Trans. on Info. Theor

MRD codes with maximum idealizers

Left and right idealizers are important invariants of linear rank-distance codes. In the case of maximum rank-distance (MRD for short) codes in $\mathbb{F}_q^{n\times n}$ the idealizers have been proved to be isomorphic to finite fields of size at most $q^n$. Up to now, the only known MRD codes with maximum left and right idealizers are generalized Gabidulin codes, which were first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and Gabidulin in 2005. In this paper we classify MRD codes in $\mathbb{F}_q^{n\times n}$ for $n\leq 9$ with maximum left and right idealizers and connect them to Moore-type matrices. Apart from generalized Gabidulin codes, it turns out that there is a further family of rank-distance codes providing MRD ones with maximum idealizers for $n=7$, $q$ odd and for $n=8$, $q\equiv 1 \pmod 3$. These codes are not equivalent to any previously known MRD code. Moreover, we show that this family of rank-distance codes does not provide any further examples for $n\geq 9$.Comment: Reviewers' comments implemented, we changed the titl

On MDS Negacyclic LCD Codes

Linear codes with complementary duals (LCD) have a great deal of significance amongst linear codes. Maximum distance separable (MDS) codes are also an important class of linear codes since they achieve the greatest error correcting and detecting capabilities for fixed length and dimension. The construction of linear codes that are both LCD and MDS is a hard task in coding theory. In this paper, we study the constructions of LCD codes that are MDS from negacyclic codes over finite fields of odd prime power $q$ elements. We construct four families of MDS negacyclic LCD codes of length $n|\frac{{q-1}}{2}$, $n|\frac{{q+1}}{2}$ and a family of negacyclic LCD codes of length $n=q-1$. Furthermore, we obtain five families of $q^{2}$-ary Hermitian MDS negacyclic LCD codes of length $n|\left( q-1\right)$ and four families of Hermitian negacyclic LCD codes of length $n=q^{2}+1.$ For both Euclidean and Hermitian cases the dimensions of these codes are determined and for some classes the minimum distances are settled. For the other cases, by studying $q$ and $q^{2}$-cyclotomic classes we give lower bounds on the minimum distance