Abstract Due to its realistic appearance, computational convenience, and efficient Monte Carlo sampling, Ward's anisotropic BRDF is widely used in computer graphics for modeling specular reflection. Incorporating the criticism that the Ward and the Ward‐Dür model do not meet energy balance at grazing angles, we propose a modified BRDF that is energy conserving and preserves Helmholtz reciprocity. The new BRDF is computationally cheap to evaluate, admits efficient importance sampling, and thus sustains the main benefits of the Ward model. We show that the proposed BRDF is better suited for fitting measured reflectance data of a linoleum floor used in a real‐world building than the Ward and the Ward‐Dür model.

Fast interpolation algorithms are presented for sparse sums of characters on the product monoid $U^n $ with values in a field K in two special cases. In the first case, U is a finite cyclic group of order e, and K is a field that contains a root of unity of order e. Here linear codes over ${\mathbb{Z} / {e\mathbb{Z}}}$ can be used to construct sets of evaluation points that allow efficient interpolation by decoding Reed–Solomon codes. In the second case, K is the binary field $GF(2)$, and U is the multiplicative monoid of $GF(2)$. Here sparse sums of characters coincide with sparse Boolean polynomials and can be interpolated using the smallest set of evaluation points by decoding Reed–Muller codes.

BCH decoders based on the Peterson-Gorenstein-Zierler algorithm can malfunction and produce output vectors that are no codewords. To avoid this malfunction a simple additional check is proposed that can be performed before the computation of the error locations. The additional check consists of the minimum number of algebraic equations in the syndrome components that are necessary over a general field to obtain a bounded-distance decoder.>

A novel fast algorithm is developed for computing the burst-correcting limit of a cyclic or shortened cyclic code from the parity-check polynomial of the cyclic code. The algorithm is similar to the algorithm of H.J. Matt and J.L. Massey (1980) which, up to now, has been the most efficient method for determining the burst-correcting limit of a cyclic code, but is based on apolarity of binary forms instead of linear complexity. The running times of implementations in C of both algorithms on an IBM RISC System/6000 are compared for several binary cyclic codes of practical interest. A table of the burst-correcting limit of primitive binary BCH codes of length up to 1023 is included.>

When a block code is used on a discrete memoryless channel with an incomplete decoding rule that is based on a generalized distance, the probability of decoding failure, the probability of erroneous decoding, and the expected number of symbol decoding errors can be expressed in terms of the generalized weight enumerator polynomials of the code. For the symmetric erasure channel, numerically stable methods to compute these probabilities or expectations are proposed for binary codes whose distance distributions are known, and for linear maximum distance separable (MDS) codes. The method for linear MDS codes saves the computation of the weight distribution and yields upper bounds for the probability of erroneous decoding and for the symbol error rate by the cumulative binomial distribution. Numerical examples include a triple-error-correcting Bose-Chaudhuri-Hocquenghem (BCH) code of length 63 and a Reed-Solomon code of length 1023 and minimum distance 31.>