Abstract. Determining deep holes is an important topic in decoding Reed-Solomon codes. Cheng and Murray, Li and Wan, Wu and Hong investigated the error distance of generalized Reed-Solomon codes. Recently, Zhang and Wan explored the deep holes of projective Reed-Solomon codes. Let l ≥ 1 be an integer and a1, . . . , al be arbitrarily given l distinct elements of the finite field Fq of q element...

4 Ideals and cyclic codes 18 4.1 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Cyclic codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 Group of a code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.4 Minimal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.5 BCH and Reed-Solomon codes . . . . . . . . . . ....

The complexity of maximum likelihood decoding of the ReedSolomon codes [q−1, k]q is a well known open problem. The only known result [4] in this direction states that it is at least as hard as the discrete logarithm in some cases where the information rate unfortunately goes to zero. In this paper, we remove the rate restriction and prove that the same complexity result holds for any positive i...

In this paper, we present a new class of public key cryptosystem based on Reed-Solomon codes, a member of the code based PKC(CBPKC), referred to as K(XII)SE(1)PKC. We show that K(XII)SE(1)PKC can be secure against the various attacks. Particularly we present a member of K(XII)SE(1)PKC constructed based on the Reed-Solomon code over the extension field F28 , which is extensively used in the pres...

Given a classical error-correcting block code, the task of quantum list decoding is to produce from any quantumly corrupted codeword a short list containing all messages whose codewords exhibit high “presence” in the quantumly corrupted codeword. Efficient quantum list decoders have been used to prove a quantum hardcore property of classical codes. However, the code rates of all known families ...

In this work the definition of codes as modules over skew polynomial rings of automorphism type is generalized to skew polynomial rings, whose multiplication is defined using an automorphism and a derivation. This produces a more general class of codes which, in some cases, produce better distance bounds than module skew codes constructed only with an automorphism. Extending the approach of Gab...

Binary low-density parity-check (LDPC) codes perform very well on magnetic recording channels (MRCs) with additive white Gaussian noise (AWGN). However, an MRC is subject to other impairments, such as media defects and thermal asperities. Binary LDPC codes may not be able to cope with these impairments without the help of a Reed–Solomon code. A better form of coding may be -ary LDPC codes, whic...

We study the distinguishability of linearized Reed–Solomon (LRS) codes by defining and analyzing analogs square-code Overbeck distinguisher for classical Gabidulin codes, respectively. Our main results show that works generalized (GLRS) defined with trivial automorphism, whereas Overbeck-type can handle LRS in general setting. further how to recover code parameters from any generator matrix suc...

The classical family of [n, k]q Reed-Solomon codes over a field Fq consist of the evaluations of polynomials f ∈ Fq[X ] of degree< k at n distinct field elements. In this work, we consider a closely related family of codes, called (orderm) derivative codes and defined over fields of large characteristic, which consist of the evaluations of f as well as its first m− 1 formal derivatives at n dis...

We construct maximally recoverable codes (corresponding to partial MDS codes) which are based on linearized Reed-Solomon codes. The new have a smaller field size requirement compared with known constructions. For certain asymptotic regimes, the constructed order-optimal alphabet size, asymptotically matching lower bound.