Luther D. Rudolph

An optimum symbol-by-symbol decoding rule for linear codes

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.

Decoding by local optimization (Corresp.)

The maximum likelihood decoding problem for linear binary (n,k) codes is reformulated as a continuous optimization problem in a k -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.

Threshold decoding of cyclic codes

It is shown that every cyclic code over GF(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 all n -tuples with elements from the ring of integers modulo p .

On the structure of generalized finite-geometry codes

Some new results on the structure of generalized finitegeometry codes are presented.

Decoding by sequential code reduction

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.

Asymptotic performance of optimum bit-by-bit decoding for the white Gaussian channel (Corresp.)

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.

One-step weighted-majority decoding (Corresp.)

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.

Iteratively maximum likelihood decodable spherical codes and a method for their construction

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. >

Generalized threshold decoding of convolutional codes

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.

On Two-Level Exclusive-or Majority Networks

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.