3,552 research outputs found
Counting generalized Reed-Solomon codes
In this article we count the number of generalized Reed-Solomon (GRS) codes
of dimension k and length n, including the codes coming from a non-degenerate
conic plus nucleus. We compare our results with known formulae for the number
of 3-dimensional MDS codes of length n=6,7,8,9
Optimal Thresholds for GMD Decoding with (L+1)/L-extended Bounded Distance Decoders
We investigate threshold-based multi-trial decoding of concatenated codes
with an inner Maximum-Likelihood decoder and an outer error/erasure
(L+1)/L-extended Bounded Distance decoder, i.e. a decoder which corrects e
errors and t erasures if e(L+1)/L + t <= d - 1, where d is the minimum distance
of the outer code and L is a positive integer. This is a generalization of
Forney's GMD decoding, which was considered only for L = 1, i.e. outer Bounded
Minimum Distance decoding. One important example for (L+1)/L-extended Bounded
Distance decoders is decoding of L-Interleaved Reed-Solomon codes. Our main
contribution is a threshold location formula, which allows to optimally erase
unreliable inner decoding results, for a given number of decoding trials and
parameter L. Thereby, the term optimal means that the residual codeword error
probability of the concatenated code is minimized. We give an estimation of
this probability for any number of decoding trials.Comment: Accepted for the 2010 IEEE International Symposium on Information
Theory, Austin, TX, USA, June 13 - 18, 2010. 5 pages, 2 figure
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
On the minimum distance of elliptic curve codes
Computing the minimum distance of a linear code is one of the fundamental
problems in algorithmic coding theory. Vardy [14] showed that it is an \np-hard
problem for general linear codes. In practice, one often uses codes with
additional mathematical structure, such as AG codes. For AG codes of genus
(generalized Reed-Solomon codes), the minimum distance has a simple explicit
formula. An interesting result of Cheng [3] says that the minimum distance
problem is already \np-hard (under \rp-reduction) for general elliptic curve
codes (ECAG codes, or AG codes of genus ). In this paper, we show that the
minimum distance of ECAG codes also has a simple explicit formula if the
evaluation set is suitably large (at least of the group order). Our
method is purely combinatorial and based on a new sieving technique from the
first two authors [8]. This method also proves a significantly stronger version
of the MDS (maximum distance separable) conjecture for ECAG codes.Comment: 13 page
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