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Register a new userTwisted Reed-Solomon Codes
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Added by
Sven Puchinger
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In this article, we present a new construction of evaluation codes in the Hamming metric, which we call twisted Reed-Solomon codes. Whereas Reed-Solomon (RS) codes are MDS codes, this need not be the case for twisted RS codes. Nonetheless, we show that our construction yields several families of MDS codes. Further, for a large subclass of (MDS) twisted RS codes, we show that the new codes are not generalized RS codes. To achieve this, we use properties of Schur squares of codes as well as an explicit description of the dual of a large subclass of our codes. We conclude the paper with a description of a decoder, that performs very well in practice as shown by extensive simulation results.
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Maximum distance separable (MDS) codes are optimal where the minimum distance cannot be improved for a given length and code size. Twisted Reed-Solomon codes over finite fields were introduced in 2017, which are generalization of Reed-Solomon codes. Twisted Reed-Solomon codes can be applied in cryptography which prefer the codes with large minimum distance. MDS codes can be constructed from twisted Reed-Solomon codes, and most of them are not equivalent to Reed-Solomon codes. In this paper, we first generalize twisted Reed-Solomon codes to generalized twisted Reed-Solomon codes, then we give some new explicit constructions of MDS (generalized) twisted Reed-Solomon codes. In some cases, our constructions can get MDS codes with the length longer than the constructions of previous works. Linear complementary dual (LCD) codes are linear codes that intersect with their duals trivially. LCD codes can be applied in cryptography. This application of LCD codes renewed the interest in the construction of LCD codes having a large minimum distance. We also provide new constructions of LCD MDS codes from generalized twisted Reed-Solomon codes.
Guo, Kopparty and Sudan have initiated the study of error-correcting codes derived by lifting of affine-invariant codes. Lifted Reed-Solomon (RS) codes are defined as the evaluation of polynomials in a vector space over a field by requiring their restriction to every line in the space to be a codeword of the RS code. In this paper, we investigate lifted RS codes and discuss their application to batch codes, a notion introduced in the context of private information retrieval and load-balancing in distributed storage systems. First, we improve the estimate of the code rate of lifted RS codes for lifting parameter $mge 3$ and large field size. Second, a new explicit construction of batch codes utilizing lifted RS codes is proposed. For some parameter regimes, our codes have a better trade-off between parameters than previously known batch codes.
A linear code is called an MDS self-dual code if it is both an MDS code and a self-dual code with respect to the Euclidean inner product. The parameters of such codes are completely determined by the code length. In this paper, we consider new constructions of MDS self-dual codes via generalized Reed-Solomon (GRS) codes and their extended codes. The critical idea of our constructions is to choose suitable evaluation points such that the corresponding (extended) GRS codes are self-dual. The evaluation set of our constructions is consists of a subgroup of finite fields and its cosets in a bigger subgroup. Four new families of MDS self-dual codes are obtained and they have better parameters than previous results in certain region. Moreover, by the Mobius action over finite fields, we give a systematic way to construct self-dual GRS codes with different evaluation points provided any known self-dual GRS codes. Specially, we prove that all the self-dual extended GRS codes over $mathbb{F}_{q}$ with length $n< q+1$ can be constructed from GRS codes with the same parameters.
Linearized Reed-Solomon (LRS) codes are sum-rank metric codes that fulfill the Singleton bound with equality. In the two extreme cases of the sum-rank metric, they coincide with Reed-Solomon codes (Hamming metric) and Gabidulin codes (rank metric). List decoding in these extreme cases is well-studied, and the two code classes behave very differently in terms of list size, but nothing is known for the general case. In this paper, we derive a lower bound on the list size for LRS codes, which is, for a large class of LRS codes, exponential directly above the Johnson radius. Furthermore, we show that some families of linearized Reed-Solomon codes with constant numbers of blocks cannot be list decoded beyond the unique decoding radius.
Two concatenated coding schemes incorporating algebraic Reed-Solomon (RS) codes and polarization-adjusted convolutional (PAC) codes are proposed. Simulation results show that at a bit error rate of $10^{-5}$, a concatenated scheme using RS and PAC codes has more than $0.25$ dB coding gain over the NASA standard concatenation scheme, which uses RS and convolutional codes.
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