The authors propose a class of spherical codes which can be easily decoded by an efficient iterative maximum likelihood decoding algorithm. A necessary and sufficient condition for a spherical code to be iteratively maximum likelihood decodable is formulated. A systematic construction method for such codes based on shrinking of Voronoi corners is analyzed. The base code used for construction is the binary maximal length sequence code. The second-level construction is described. Computer simulation results for selected codes constructed by the proposed method are given.>

The maximum likelihood decoding problem for linear binary(n,k)codes is reformulated as a continuous optimization problem in ak-dimensional solid cube. We obtain a near optimum solution of this problem by use of a simple gradient local optimization algorithm. Computer simulation results are presented for the(21,11)projective geometry code and the(47,23)quadratic-residue code.

This correspondence presents a oncept of function-independent testing of digital networks. It is based on the idea of dual-mode logic where the network is tested in one mode while the normal function of the network is performed in another mode, with neither mode interfering with the other. This correspondence simultaneously defines the structure of modules with the above characteristics such that combinational and sequential networks built with them can be tested with two and six function-independent tests, respectively.

Bit-by-bit soft-decision decoding of binary cyclic codes is considered. A significant reduction in decoder complexity can be achieved by requiring only that the decoder correct all analog error patterns which fall within a Euclidean sphere whose radius is equal to half the minimum Euclidean distance of the code. Such a "maximum-radius" scheme is asymptotically optimum for the additive white Gaussian noise (AWGN) channel. An iterative extension of the basic algebraic analog decoding scheme is discussed, and performance curves are given for the (17,9), (21,11), and (73,45) codes on the AWGN channel.

Asymptotic expressions are derived for the probability of bit error for optimum bit-by-bit decoding of a linear binary block code for the white Gaussian noise channel.

A decoding rule is presented which minimizes the probability of symbol error over a time-discrete memory]ess channel for any linear error-correcting code when the codewords are equiprobable. The complexity of this rule varies inversely with code rate, making the technique particularly attractive for high rate codes. Examples are given for both block and convolutional codes.

It is shown that not all Boolean functions can be realized by a two-level EXCLUSIVE-OR majority network. However, if repeats at the first level are allowed, then it is shown that such a network is universal. A minimal weight vector with respect to this latter network is defined. By using the restricted-affine-group (RAG) equivalence of Boolean functions, it is shown that if two functions are in the same RAG class, then they are realized by the same minimal weight vector to within permutations and/or sign changes.

Some new results on the structure of generalized finite-geometry codes are presented.

This paper presents majority-decoding algorithms for four classes of binary cyclic codes. The classes are those for which the parity-check polynomial is: 1) the product of two primitive polynomials with relatively prime exponents and 2) the product of(x^r + 1)/(x + 1)and a primitive polynomial, wherer \geq 3is odd and the exponents are relatively prime, together with the corresponding nonexpurgated codes.

A general decoding method for cyclic codes is presented which gives promise of substantially reducing the complexity of decoders at the cost of a modest increase in decoding time (or delay). Significant reductions in decoder complexity for binary cyclic finite-geometry codes are demonstrated.

A bound is derived on the number of low-weight sequences an L-step observable nonsingular-autonomous finite-state machine is capable of producing.

It is shown that any decoding function for a linear binary code can be realized as a weighted majority of nonorthogonal parity checks. An example is given of a four-error-correcting code that is neither L-step orthogonalizable nor one-step majority decodable using non-orthogonal parity checks and yet is one-step weighted-majority decodable using only ten nonorthogonal parity checks.

A counterexample to a conjecture on the number of constraint lengths required to achieve the free distance of a rate1/nsystematic convolutional code is presented.

A one-step threshold decoding method previously presented for cyclic block codes is shown to apply generally to linear convolutional codes. It is further shown that this method generalizes in a natural way to allow decoding of the received sequence in its unquantized analog form.

It is shown that every cyclic code overGF(p)can be decoded up to its minimum distance by a threshold decoder employing general parity checks and a single threshold element. This result is obtained through the application of a general decomposition theorem for complex-valued functions defined on the space of alln-tuples with elements from the ring of integers modulop.