The Dimension of Subcode-Subfields of Shortened Generalized Reed Solomon Codes

Reed-Solomon (RS) codes are among the most ubiquitous codes due to their good parameters as well as efficient encoding and decoding procedures. However, RS codes suffer from having a fixed length. In many applications where the length is static, the appropriate length can be obtained by an RS code by shortening or puncturing. Generalized Reed-Solomon (GRS) codes are a generalization of RS codes, whose subfield-subcodes are extensively studied. In this paper we show that a particular class of GRS codes produces many subfield-subcodes with large dimension. An algorithm for searching through the codes is presented as well as a list of new codes obtained from this method

Subfield-Subcodes of Generalized Toric codes

We study subfield-subcodes of Generalized Toric (GT) codes over $\mathbb{F}_{p^s}$. These are the multidimensional analogues of BCH codes, which may be seen as subfield-subcodes of generalized Reed-Solomon codes. We identify polynomial generators for subfield-subcodes of GT codes which allows us to determine the dimensions and obtain bounds for the minimum distance. We give several examples of binary and ternary subfield-subcodes of GT codes that are the best known codes of a given dimension and length.Comment: Submitted to 2010 IEEE International Symposium on Information Theory (ISIT 2010

Subspace subcodes of Reed-Solomon codes

We introduce a class of nonlinear cyclic error-correcting codes, which we call subspace subcodes of Reed-Solomon (SSRS) codes. An SSRS code is a subset of a parent Reed-Solomon (RS) code consisting of the RS codewords whose components all lie in a fixed ν-dimensional vector subspace S of GF (2m). SSRS codes are constructed using properties of the Galois field GF(2m). They are not linear over the field GF(2ν), which does not come into play, but rather are Abelian group codes over S. However, they are linear over GF(2), and the symbol-wise cyclic shift of any codeword is also a codeword. Our main result is an explicit but complicated formula for the dimension of an SSRS code. It implies a simple lower bound, which gives the true value of the dimension for most, though not all, subspaces. We also prove several important duality properties. We present some numerical examples, which show, among other things, that (1) SSRS codes can have a higher dimension than comparable subfield subcodes of RS codes, so that even if GF(2ν) is a subfield of GF(2m), it may not be the best ν-dimensional subspace for constructing SSRS codes; and (2) many high-rate SSRS codes have a larger dimension than any previously known code with the same values of n, d, and q, including algebraic-geometry codes. These examples suggest that high-rate SSRS codes are promising candidates to replace Reed-Solomon codes in high-performance transmission and storage systems

New Identities Relating Wild Goppa Codes

For a given support $L \in \mathbb{F}_{q^m}^n$ and a polynomial $g\in \mathbb{F}_{q^m}[x]$ with no roots in $\mathbb{F}_{q^m}$, we prove equality between the $q$-ary Goppa codes $\Gamma_q(L,N(g)) = \Gamma_q(L,N(g)/g)$ where $N(g)$ denotes the norm of $g$, that is $g^{q^{m-1}+\cdots +q+1}.$ In particular, for $m=2$, that is, for a quadratic extension, we get $\Gamma_q(L,g^q) = \Gamma_q(L,g^{q+1})$. If $g$ has roots in $\mathbb{F}_{q^m}$, then we do not necessarily have equality and we prove that the difference of the dimensions of the two codes is bounded above by the number of distinct roots of $g$ in $\mathbb{F}_{q^m}$. These identities provide numerous code equivalences and improved designed parameters for some families of classical Goppa codes.Comment: 14 page

Cyclic LRC Codes, binary LRC codes, and upper bounds on the distance of cyclic codes

We consider linear cyclic codes with the locality property, or locally recoverable codes (LRC codes). A family of LRC codes that generalize the classical construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Trans. Inform. Theory, no. 8, 2014). In this paper we focus on optimal cyclic codes that arise from this construction. We give a characterization of these codes in terms of their zeros, and observe that there are many equivalent ways of constructing optimal cyclic LRC codes over a given field. We also study subfield subcodes of cyclic LRC codes (BCH-like LRC codes) and establish several results about their locality and minimum distance. The locality parameter of a cyclic code is related to the dual distance of this code, and we phrase our results in terms of upper bounds on the dual distance.Comment: 12pp., submitted for publication. An extended abstract of this submission was posted earlier as arXiv:1502.01414 and was published in Proceedings of the 2015 IEEE International Symposium on Information Theory, Hong Kong, China, June 14-19, 2015, pp. 1262--126

New Quantum Codes from Evaluation and Matrix-Product Codes

Stabilizer codes obtained via CSS code construction and Steane's enlargement of subfield-subcodes and matrix-product codes coming from generalized Reed-Muller, hyperbolic and affine variety codes are studied. Stabilizer codes with good quantum parameters are supplied, in particular, some binary codes of lengths 127 and 128 improve the parameters of the codes in http://www.codetables.de. Moreover, non-binary codes are presented either with parameters better than or equal to the quantum codes obtained from BCH codes by La Guardia or with lengths that can not be reached by them