In this paper, we consider a class of exponential sums from some half quadratic binomials. The exponential sums are proven to be eleven-valued with maximal magnitude (1/2(q - √q)) except for the trivial value q. As applications, first, we investigate the autocorrelation and cross-correlation distribution among the sequences in a sequence family. Second, we determine the weight distributions of several classes of linear codes. Some of the dual codes have minimum distance four, which are optimal with respect to the Hamming bound. Our results extend the result by Choi et al. and show that some correlation values in it do not occur.

In this paper, a class of exponential sum from some binomials is studied. As an application, we determine the weight distribution of a family of p-ary cyclic codes.

In this paper, the large family of generalized Kasami sequences has been studied. In particular, the cross-correlation distribution among these sequences has been explicitly calculated. Meanwhile, the weight distributions of two classes of cyclic codes could also be determined. This paper generalizes the results from several previous papers.

Let q = pnwith n = 2m and p be a prime. Let 0 ≤ k ≤ n − 1, k ≠ m. Assume k/(m,k) and m/(m,k) are both odd. In this paper we will study the following exponential sums of the equation where Tr1n: Fq− Fpand Tr1m: Fpm→ Fpare the canonical trace mappings and ζp= e 2πi/p is a primitive p-th root of unity. As an application, we will determine the weight distribution of the cyclic codes C over Fpwith parity-check polynomial h1(x)h2(x) where h1(x) and h2(x) are the minimal polynomials of πequation and πequation over Fprespectively for a primitive element π of Fq.

For the decoding of a binary linear block code of minimal Hamming distancedover additive white Gaussian noise (AWGN) channels, a soft-decision decoder achieves bounded-distance (BD) decoding if its squared error-correction radius is equal tod. A Chase-like algorithm outputs the best (most likely) codeword in a list of candidates generated by a conventional algebraic binary decoder in a few trials. It is of interest to design Chase-like algorithms that achieve BD decoding with as least trials as possible. In this paper, we show that Chase-like algorithms can achieve BD decoding with onlyO(d1/2+epsiv) trials for any given positive numberepsiv.

In this correspondence, we consider the decoding of binary block codes over the additive white Gaussian noise (AWGN) channel with binary phase-shift keying (BPSK) signaling. By a reliability-order-based decoding algorithm (ROBDA), we mean a soft-decision decoding algorithm which decodes to the best (most likely) codeword of the form that is the sum of the hard-decision tuple and an error pattern in a set determined only by the order of the reliabilities of the hard decisions. Examples of ROBDAs include many well-known decoding algorithms, such as the generalized-minimum-distance (GMD) decoding algorithm, Chase decoding algorithms, and the reliability-based decoding algorithms proposed by Fossorier and Lin. It is known that the squared error-correction-radii of ROBDAs can be computed from the minimal squared Euclidean distances (MSEDs) between the all-one sequence and the polyhedra corresponding to the error patterns. For the computation of such MSEDs, we give a new method which is more compact than the one proposed by Fossorier and Lin. These results are further used to show that any bounded-distance ROBDA is asymptotically optimal: The ratio between the probability of decoding error of a bounded-distance ROBDA and that of the maximum-likelihood (ML) decoding approaches 1 when the signal-to-noise ratio (SNR) approaches infinity, provided that the minimum Hamming distance of the code is greater than 2.

The generalized minimum distance (GMD) and Chase (1972) decoding algorithms are some of the most important suboptimum bounded distance decoding algorithms for binary linear block codes over an additive white Gaussian noise (AWGN) channel. We compute the limitation of the ratio between the probability of decoding error for the GMD or any one of the Chase decoding algorithms and that of the maximum-likelihood (ML) decoding when the signal-to-noise ratio (SNR) approaches infinity. If the minimum Hamming distance of the code is greater than 2, the limitation is shown to be equal to 1 and thus the GMD and Chase decoding algorithms are asymptotically optimum.