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Register a new userMinimizing costs of communication with random constant weight codes
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Added by
Pau Vilimelis Aceituno
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Authors
Pau Vilimelis Aceituno
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No Arabic abstract
We present a framework for minimizing costs in constant weight codes while maintaining a certain amount of differentiable codewords. Our calculations are based on a combinatorial view of constant weight codes and relay on simple approximations.
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This paper focuses on error-correcting codes that can handle a predefined set of specific error patterns. The need for such codes arises in many settings of practical interest, including wireless communication and flash memory systems. In many such settings, a smaller field size is achievable than that offered by MDS and other standard codes. We establish a connection between the minimum alphabet size for this generalized setting and the combinatorial properties of a hypergraph that represents the prespecified collection of error patterns. We also show a connection between error and erasure correcting codes in this specialized setting. This allows us to establish bounds on the minimum alphabet size and show an advantage of non-linear codes over linear codes in a generalized setting. We also consider a variation of the problem which allows a small probability of decoding error and relate it to an approximate version of hypergraph coloring.
In this paper we consider the problem of determining the weight spectrum of q-ary codes C(3,m) associated with Grassmann varieties G(3,m). For m=6 this was done by Nogin. We derive a formula for the weight of a codeword of C(3,m), in terms of certain varieties associated with alternating trilinear forms on (F_q)^m. The classification of such forms under the action of the general linear group GL(m,F_q) is the other component that is required to calculate the spectrum of C(3,m). For m=7, we explicitly determine the varieties mentioned above. The classification problem for alternating 3-forms on (F_q)^7 was solved by Cohen and Helminck, which we then use to determine the spectrum of C(3,7).
Usually, it is difficult to determine the weight distribution of an irreducible cyclic code. In this paper, we discuss the case when an irreducible cyclic code has the maximal number of distinct nonzero weights and give a necessary and sufficient condition. In this case, we also obtain a divisible property for the weight of a codeword. Further, we present a necessary and sufficient condition for an irreducible cyclic code with only one nonzero weight. Finally, we determine the weight distribution of an irreducible cyclic code for some cases.
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In 1997, Shor and Laflamme defined the weight enumerators for quantum error-correcting codes and derived a MacWilliams identity. We extend their work by introducing our double weight enumerators and complete weight enumerators. The MacWilliams identities for these enumerators can be obtained similarly. With the help of MacWilliams identities, we obtain various bounds for asymmetric quantum codes.
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