The binary weight distribution of the extended (2 sup m, 2 sup m-4) code of Reed-Solomon code over GF(2 sup m) with generator polynomial (x-alpha sup 2) (x-alpha sup 3)

Consider an (n,k) linear code with symbols from GF(2 sup m). If each code symbol is represented by a binary m-tuple using a certain basis for GF(2 sup m), a binary (nm,km) linear code called a binary image of the original code is obtained. A lower bound is presented on the minimum weight enumerator for a binary image of the extended (2 sup m, 2 sup m -4) code of Reed-Solomon code over GF(2 sup m) with generator polynomical (x - alpha)(x- alpha squared)(x - alpha cubed) and its dual code, where alpha is a primitive element in GF(2 sup m)

On complexity of trellis structure of linear block codes

The trellis structure of linear block codes (LBCs) is discussed. The state and branch complexities of a trellis diagram (TD) for a LBC is investigated. The TD with the minimum number of states is said to be minimal. The branch complexity of a minimal TD for a LBC is expressed in terms of the dimensions of specific subcodes of the given code. Then upper and lower bounds are derived on the number of states of a minimal TD for a LBC, and it is shown that a cyclic (or shortened cyclic) code is the worst in terms of the state complexity among the LBCs of the same length and dimension. Furthermore, it is shown that the structural complexity of a minimal TD for a LBC depends on the order of its bit positions. This fact suggests that an appropriate permutation of the bit positions of a code may result in an equivalent code with a much simpler minimal TD. Boolean polynomial representation of codewords of a LBC is also considered. This representation helps in study of the trellis structure of the code. Boolean polynomial representation of a code is applied to construct its minimal TD. Particularly, the construction of minimal trellises for Reed-Muller codes and the extended and permuted binary primitive BCH codes which contain Reed-Muller as subcodes is emphasized. Finally, the structural complexity of minimal trellises for the extended and permuted, and double-error-correcting BCH codes is analyzed and presented. It is shown that these codes have relatively simple trellis structure and hence can be decoded with the Viterbi decoding algorithm

A concatenated coded modulation scheme for error control

A concatenated coded modulation scheme for error control in data communications is presented. The scheme is achieved by concatenating a Reed-Solomon outer code and a bandwidth efficient block inner code for M-ary PSK modulation. Error performance of the scheme is analyzed for an AWGN channel. It is shown that extremely high reliability can be attained by using a simple M-ary PSK modulation inner code and a relatively powerful Reed-Solomon outer code. Furthermore, if an inner code of high effective rate is used, the bandwidth expansion required by the scheme due to coding will be greatly reduced. The proposed scheme is very effective for high speed satellite communications for large file transfer where high reliability is required. A simple method is also presented for constructing codes for M-ary PSK modulation. Some short M-ary PSK codes with good minimum squared Euclidean distance are constructed. These codes have trellis structure and hence can be decoded with a soft decision Viterbi decoding algorithm. Furthermore, some of these codes are phase invariant under multiples of 45 deg rotation

On linear structure and phase rotation invariant properties of block 2(sup l)-PSK modulation codes

Two important structural properties of block 2(l)-ary PSK (phase shift keying) modulation codes, linear structure and phase symmetry, are investigated. For an additive white Gaussian noise (AWGN) channel, the error performance of a modulation code depends on its squared Euclidean distance distribution. Linear structure of a code makes the error performance analysis much easier. Phase symmetry of a code is important in resolving carrier phase ambiguity and ensuring rapid carrier phase resynchronization after temporary loss of synchronization. It is desirable for a code to have as many phase symmetries as possible. A 2(l)-ary modulation code is represented here as a code with symbols from the integer group. S sub 2(l) PSK = (0,1,2,...,2(l)-1), under the modulo-2(l) addition. The linear structure of block 2(l)-ary PSK modulation codes over S sub 2(l)-ary PSK with respect to the modulo-2(l) vector addition is defined, and conditions under which a block 2(l)-ary PSK modulation code is linear are derived. Once the linear structure is developed, phase symmetry of a block 2(l)-ary PSK modulation code is studied. It is a necessary and sufficient condition for a block 2(l)-PSK modulation code, which is linear as a binary code, to be invariant under 180 deg/2(l-h) phase rotation, for 1 is less than or equal to h is less than or equal to l. A list of short 8-PSK and 16-PSK modulation codes is given, together with their linear structure and the smallest phase rotation for which a code is invariant

Serial-parallel multiplication in Galois fields

A method for multiplying two elements from the Galois field GF(2 sup ms) is presented. This method provides a tradeoff between speed and complexity