Self-dual block codes of rate 1/2 are constructed. The codes are of length 8m with weights w, w=0 mod 4. The codes have a convolutional portion of length 8m-2 and non-systematic information length 4m-1. The last two bits are parity checks on the two (4m-1) length parity sequences. The final information bit complements one of the extended parity sequences of length 4m. Solomon and Van Tilborg (1979) have developed algorithms to generate these for the quadratic residue codes of lengths 24 and beyond. For reasonable constraint lengths, there are possible Viterbi type of decodings that may be as simple as in the convolutional encoding/decoding of the extended Golay code. In addition, the K=9 constraint length for the QR (48, 24;12) code found by Solomon and Van Tilborg is lowered to K=8.>
Self-dual block codes of rate 1/2 are constructed. The codes are of length 8m with weights w, w=0 mod 4. The codes have a convolutional portion of length 8m-2 and non-systematic information length 4m-1. The last two bits are parity checks on the two (4m-1) length parity sequences. The final information bit complements one of the extended parity sequences of length 4m. Solomon and Van Tilborg (1979) have developed algorithms to generate these for the quadratic residue codes of lengths 24 and beyond. For reasonable constraint lengths, there are possible Viterbi type of decodings that may be as simple as in the convolutional encoding/decoding of the extended Golay code. In addition, the K=9 constraint length for the QR (48, 24;12) code found by Solomon and Van Tilborg is lowered to K=8.>
There exist almost complete convolutional encodings of self-dual codes, i.e., block codes of rate 1/2 with weights w, w = 0 mod 4. The codes are of length 8m with the convolutional portion of length 8m-2 and the nonsystematic information of length 4m-1. The last two bits are parity checks on the two (4m-1) length parity sequences. The final information bit complements one of the extended parity sequences of length 4m. Solomon and van Tilborg have developed algorithms to generate these for the Quadratic Residue (QR) Codes of lengths 48 and beyond. For these codes and reasonable constraint lengths, there are sequential decodings for both hard and soft decisions. There are also possible Viterbi-type decodings that may be simple, as in a convolutional encoding/decoding of the extended Golay Code. In addition, the previously found constraint length K = 9 for the QR (48, 24;12) Code is lowered here to K = 8.
In this paper we obtain the [60,30,12], [64,32,12], [68,34,12], [72,36,12] self-dual codes as tailbitting convolutional codes with the smallest constraint length K=9. In this construction one information bit is modulo two added to the one of the encoder outputs and the first row in the quasi-cyclic generator matrix is replaced by the obtained row. The pure quasi-cyclic construction with K=10 is also available for [68,34,12] and [72,36,12] codes. The new [72,36,12] singly even self-dual code with parameters Beta=483 Gamma=0 was obtained.
The Pless construction of the (24, 12, 8) extended Golay code can be modified to construct the (32, 16, 8) extended QR code. Some structural similarities are discovered between the (32, 16, 8) QR code and the (24, 12, 8) Golay code. A simple decoder is developed based on the two-level structure of the (32, 16, 8) QR code. In addition to correcting all three or less bit errors, the proposed algorithm is able to correct some four-bit error patterns. Recently, soft-logic decoding of block codes has found increasing practical applications. The two-level structure renders an extension of Vardy and Be‘ery’s fast algorithm to the (32, 16, 8) QR code. The Vardy-Be‘ery’s algorithm is, so far, the most efficient scheme for the Golay code in the literature, with direct application impact. Its extension to the (32, 16, 8) code is also computationally efficient, as analysed in detail.
A 16-state tail-biting trellis for the binary (24,12,8) Golay code C/sub 24/ is exhibited. This tail-biting trellis is minimal. It has many other nice properties, and may be simpler to decode than the minimal conventional trellis for C/sub 24/. Unwrapping this tail-biting trellis produces a periodically-time-varying 16-state rate- 1/2 convolutional code. This "Golay convolutional code" is the first known example of a self-dual doubly-even binary linear convolutional code. It has greater minimum distance than any 16-state time-invariant convolutional code of rate 1/2 . Demonstrably minimal tail-biting trellises are given for several other binary and nonbinary self-dual codes. For some nonbinary codes, tail-biting trellises based on code generators over a group are simpler than any tail-biting trellis based on linear generators over a field.
Four classes of maximum-girth geometrically structured column-weight-two regular quasi-cyclic (QC) low-density parity-check (LDPC) codes are introduced. Two classes of these codes, referred to as Type-I and Type-II codes, are with row-weights 4 and 3, and maximum girths 16 and 24, respectively. The idea behind the construction of these two classes of codes, with rates at least 1/2 and 1/3, is slightly generalized to obtain two classes of variable-high-rate codes, referred to as Type-III1 and Type-III2 codes, with maximum girth 20 and 16, respectively. A low-complexity deterministic algorithm for constructing these four classes of codes is given. The algorithm generates maximum-girth Type-I and Type-II codes with almost arbitrary length eta not less than 216 and 243, respectively. The output of the algorithm substantially improves on some of the previously best known codes constructed using a randomized progressive edge-growth (RPEG) algorithm. For instance, we have rate-0.71 Type-III1 codes of lengths 308 and 728 with girths 10 and 12, respectively, versus the code lengths 385 and 840 obtained by the RPEG algorithm. Simulation results on AWGN channel confirm that, from BER performance perspective, the constructed LDPC codes are superior to the column-weight-two LDPC codes constructed by the previously reported methods. The generator matrix G(D) of the convolutional codes associated with Type-I and Type-II codes is given. The free distance dfreeof such a convolutional code is equal to the minimum distance of the corresponding QC block code.