998 research outputs found
Cyclotomic trace codes
A generalization of Ding’s construction is proposed that employs as a defining set the collection of the sth powers (s ≥ 2) of all nonzero elements in GF(pm), where p ≥ 2 is prime. Some of the resulting codes are optimal or near-optimal and include projective codes over GF(4) that give rise to optimal or near optimal quantum codes. In addition, the codes yield interesting combinatorial structures, such as strongly regular graphs and block designs
A class of narrow-sense BCH codes over of length
BCH codes with efficient encoding and decoding algorithms have many
applications in communications, cryptography and combinatorics design. This
paper studies a class of linear codes of length over
with special trace representation, where is an odd prime
power. With the help of the inner distributions of some subsets of association
schemes from bilinear forms associated with quadratic forms, we determine the
weight enumerators of these codes. From determining some cyclotomic coset
leaders of cyclotomic cosets modulo , we prove
that narrow-sense BCH codes of length with designed distance
have the corresponding trace representation, and have the
minimal distance and the Bose distance , where
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
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
Quasi-cyclic subcodes of cyclic codes
We completely characterize possible indices of quasi-cyclic subcodes in a
cyclic code for a very broad class of cyclic codes. We present enumeration
results for quasi-cyclic subcodes of a fixed index and show that the problem of
enumeration is equivalent to enumeration of certain vector subspaces in finite
fields. In particular, we present enumeration results for quasi-cyclic subcodes
of the simplex code and duals of certain BCH codes. Our results are based on
the trace representation of cyclic codes
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