We consider the binary images of (8, 5) shortened cyclic codes. The (8, 5) shortened cyclic codes have a variety of choices. We have generated about 30000 sample codes with different weight distributions. Let S/sub w/ denote the set of generated sample codes with minimum weight w. The largest minimum weight of sample codes is 8. Let A/sub w/ denote the number of codewords of weight w of a sample code. In S/sub 7/, the smallest of A/sub 7/ is 10 for 10 sample codes and the second smallest of A/sub 7/ is 11 for six sample codes. In S/sub 7/, the smallest of A/sub 8/ is 728 for a sample code and the second smallest of A/sub 8/ is 729 for a sample code. We have chosen two sample codes from each of S/sub 7/ and S/sub 8/ which have the smallest and the second smallest sums of A/sub w/ for 7/spl les/w/spl les/9 in S/sub 7/ and S/sub 8/, respectively. For the AWGN channel using BPSK signaling, we have made simulation to evaluate the decoding error probabilities by a soft-decision decoding based on ordered statistics for the chosen four sample codes at SNR 2.0 to 5.0. These error probabilities are considerably smaller than the optimum error probabilities for (64, 40) subcodes of (64, 42) Reed-Muller code.
We consider the binary images of (8, 5) shortened cyclic codes. The (8, 5) shortened cyclic codes have a variety of choices. We have generated about 30000 sample codes with different weight distributions. Let S/sub w/ denote the set of generated sample codes with minimum weight w. The largest minimum weight of sample codes is 8. Let A/sub w/ denote the number of codewords of weight w of a sample code. In S/sub 7/, the smallest of A/sub 7/ is 10 for 10 sample codes and the second smallest of A/sub 7/ is 11 for six sample codes. In S/sub 7/, the smallest of A/sub 8/ is 728 for a sample code and the second smallest of A/sub 8/ is 729 for a sample code. We have chosen two sample codes from each of S/sub 7/ and S/sub 8/ which have the smallest and the second smallest sums of A/sub w/ for 7/spl les/w/spl les/9 in S/sub 7/ and S/sub 8/, respectively. For the AWGN channel using BPSK signaling, we have made simulation to evaluate the decoding error probabilities by a soft-decision decoding based on ordered statistics for the chosen four sample codes at SNR 2.0 to 5.0. These error probabilities are considerably smaller than the optimum error probabilities for (64, 40) subcodes of (64, 42) Reed-Muller code.
We present a new soft-decision majority decoding algorithm for Reed-Muller codes RM(r,m). First, the reliabilities of 2/sup m/ transmitted symbols are recalculated into the reliabilities of 2/sup m-r/ parity checks that represent each information bit. In turn, information bits are obtained by the weighted majority that gives more weight to more reliable parity checks. It is proven that for long low-rate codes RM(r,m), our soft-decision algorithm outperforms its conventional hard-decision counterpart by 10 log/sub 10/(/spl pi//2)/spl ap/2 dB at any given output error probability. For fixed code rate R and m/spl rarr//spl infin/, our algorithm increases almost 2/sup r/2/ times the correcting capability of soft-decision bounded distance decoding.
We present novel decoding schemes for hard and soft decision decoding of block codes using the minimal weight codewords of the dual code. The decoding schemes will be described for cyclic codes where polynomials can be used, however, the modification for non-cyclic codes is possible and straight forward. The hard decision decoding calculates syndrome polynomials which are the product of the received polynomial with dual codewords. Proper cyclic shifts of these syndrome polynomials are obtained and the non-zero positions are counted componentwise for these shifts. The values of this counting are a reliability measure and can be used for locating the error and also the non-error positions. This reliability measure is the basis for various variants of hard decision decoding algorithms. Decoding schemes with iterative error reduction are possible as well as information set decoding using the inherent reliability information of the measure even if there is no reliability information from the channel. Further, we will show how reliability information from the channel can be included in order to obtain soft decision decoding schemes. We derive the relation between bit flipping, believe propagation, and majority logic decoding to the novel schemes. As examples to illustrate the functioning we use BCH and Reed-Muller codes as examples for binary codes, and RS codes for non-binary codes. Besides the Plotkin construction we recall a known result that Reed-Muller codes punctured by one position are cyclic and thus, are equivalent to special cases of BCH codes. Simulation results for hard and soft decision decoding will be given for several examples and compared with results from literature. Finally, we analyze the soft decision decoding of the Plotkin construction and derive that one of the two codes uses a 3 dB better channel (also known as channel polarization).
We derive the average symbol and Hamming weight spectrum functions of the random ensemble of regular low-density parity-check (LDPC) codes over GF(2/sup b/) when used with the binary-input noisy channel. This work confirms theoretically that the near-Shannon-limit performance of Gallager's binary LDPC codes can be significantly enhanced by moving to fields of higher order. We construct a family of error-correcting codes based on the binary representation of GF(2/sup b/) codes defined on a cycle Tanner graph that appears to be "good" for both optimum and iterative decoding over the binary-input noisy channel. In particular, we report a short-block-length (1008 bits), rate-1/2 progressive-edge-growth-based cycle Tanner-graph code over GF(2/sup b/) with a block-error rate <10/sup -4/ at E/sub b//N/sub 0/=1.89 dB, which appears to exhibit the best iterative-decoding performance at this short block length known to date.
Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes, such as cyclic codes, Reed–Solomon codes, and Reed–Muller codes, have nice decoding algorithms. However, many optimal linear codes do not have an efficient decoding algorithm except for the general syndrome decoding which requires a lot of memory. Therefore, a natural question to ask is which optimal linear codes have an efficient decoding. We show that two binary optimal [ 36 , 19 , 8 ] linear codes and two binary optimal [ 40 , 22 , 8 ] codes have an efficient decoding algorithm. There was no known efficient decoding algorithm for the binary optimal [ 36 , 19 , 8 ] and [ 40 , 22 , 8 ] codes. We project them onto the much shorter length linear [ 9 , 5 , 4 ] and [ 10 , 6 , 4 ] codes over G F ( 4 ) , respectively. This decoding algorithm, called projection decoding, can correct errors of weight up to 3. These [ 36 , 19 , 8 ] and [ 40 , 22 , 8 ] codes respectively have more codewords than any optimal self-dual [ 36 , 18 , 8 ] and [ 40 , 20 , 8 ] codes for given length and minimum weight, implying that these codes are more practical.
Recursive decoding techniques are considered for Reed-Muller (RM) codes of growing length n and fixed order r. An algorithm is designed that has complexity of order nlogn and corrects most error patterns of weight up to n(1/2-/spl epsiv/) given that /spl epsiv/ exceeds n/sup -1/2r/. This improves the asymptotic bounds known for decoding RM codes with nonexponential complexity. To evaluate decoding capability, we develop a probabilistic technique that disintegrates decoding into a sequence of recursive steps. Although dependent, subsequent outputs can be tightly evaluated under the assumption that all preceding decodings are correct. In turn, this allows us to employ second-order analysis and find the error weights for which the decoding error probability vanishes on the entire sequence of decoding steps as the code length n grows.
Let E/sub m/ denote the binary (m, m-1, 2) even weight code and /spl ominus/ the Kronecker product operation. A linear code of length m/spl times/t containing E/sub m//spl ominus/(t,1,t) can be decoded efficiently using a Tanner graph and a trellis diagram together with the Wagner decoding algorithm. Thus, it is important to determine whether a given code contains such a subcode. We show that the (18,9,6) binary quadratic residue code does not contain E/sub 6//spl ominus/(3,1,3). The maximum minimum distance of such a code is 5.