62 research outputs found
The coset weight distributions of certain BCH codes and a family of curves
We study the distribution of the number of rational points in a family of
curves over a finite field of characteristic 2. This distribution determines
the coset weight distribution of a certain BCH code.Comment: Plain Tex, 15 pages; some numerical data adde
Quantum Error Correction via Codes over GF(4)
The problem of finding quantum error-correcting codes is transformed into the
problem of finding additive codes over the field GF(4) which are
self-orthogonal with respect to a certain trace inner product. Many new codes
and new bounds are presented, as well as a table of upper and lower bounds on
such codes of length up to 30 qubits.Comment: Latex, 46 pages. To appear in IEEE Transactions on Information
Theory. Replaced Sept. 24, 1996, to correct a number of minor errors.
Replaced Sept. 10, 1997. The second section has been completely rewritten,
and should hopefully be much clearer. We have also added a new section
discussing the developments of the past year. Finally, we again corrected a
number of minor error
A new efficient way based on special stabilizer multiplier permutations to attack the hardness of the minimum weight search problem for large BCH codes
BCH codes represent an important class of cyclic error-correcting codes; their minimum distances are known only for some cases and remains an open NP-Hard problem in coding theory especially for large lengths. This paper presents an efficient scheme ZSSMP (Zimmermann Special Stabilizer Multiplier Permutation) to find the true value of the minimum distance for many large BCH codes. The proposed method consists in searching a codeword having the minimum weight by Zimmermann algorithm in the sub codes fixed by special stabilizer multiplier permutations. These few sub codes had very small dimensions compared to the dimension of the considered code itself and therefore the search of a codeword of global minimum weight is simplified in terms of run time complexity. ZSSMP is validated on all BCH codes of length 255 for which it gives the exact value of the minimum distance. For BCH codes of length 511, the proposed technique passes considerably the famous known powerful scheme of Canteaut and Chabaud used to attack the public-key cryptosystems based on codes. ZSSMP is very rapid and allows catching the smallest weight codewords in few seconds. By exploiting the efficiency and the quickness of ZSSMP, the true minimum distances and consequently the error correcting capability of all the set of 165 BCH codes of length up to 1023 are determined except the two cases of the BCH(511,148) and BCH(511,259) codes. The comparison of ZSSMP with other powerful methods proves its quality for attacking the hardness of minimum weight search problem at least for the codes studied in this paper
PERFORMANCE COMPARISON OF NON-INTERLEAVED BCH CODES AND INTERLEAVED BCH CODES
This project covers the research about the BCH error correcting codes and the
performance of interleaved and non-interleaved BCH codes. Both long and short
BCH codes for multimedia communication are examined in an A WGN channel.
Algorithm for simulating the BCH codes was also being investigated, which includes
generating the parity check matrix, generating the message code in Galois array
matrix, encoding the message blocks, modulation and decoding the message blocks.
Algorithm for interleaving that includes interleaving message, including burst errors
and deinterleaving message is combined with the BCH codes algorithm for
simulating the interleaved BCH codes. The performance and feasibility of the coding
structure are tested. The performance comparison between interleaved and noninterleaved
BCH codes is studied in terms of error performance, channel performance
and effect of data rates on the bit error rate (BER). The Berlekamp-Massey Algorithm
decoding scheme was implemented. Random integers are generated and encoded with
BCH encoder. Burst errors are added before the message is interleaved, then enter
modulation and channel simulation. Interleaved message is then compared with noninterleaved
message and the error statistics are compared. Initially, certain amount of
burst errors is used. "ft is found that the graph does not agree with the theoretical bit
error rate (BER) versus signal-to-noise ratio (SNR). When compared between each
BCH codeword (i.e. n = 31, n = 63 and n = 127), n = 31 shows the highest BER while
n = 127 shows the lowest BER. This happened because of the occurrence of error
bursts and also due to error frequency. A reduced size or errors from previous is used
in the algorithm. A graph similar to the theoretical BER vs SNR is obtained for both
interleaved and non-interleaved BCH codes. It is found that BER of non-interleaved
is higher than interleaved BCH codes as SNR increases. These observations show that
size of errors influence the effect of interleaving. Simulation time is also studied in
terms of block length. It is found that interleaved BCH codes consume longer
simulation time compared to non-interleaved BCH codes due to additional algorithm
for the interleaved BCH codes
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