We use shortened and punctured codes to give an elementary proof of a combinatorial identity of Brualdi, Pless, and Beissinger from which the MacWilliams identities follow as special cases. We also give a short, mostl...
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We use shortened and punctured codes to give an elementary proof of a combinatorial identity of Brualdi, Pless, and Beissinger from which the MacWilliams identities follow as special cases. We also give a short, mostly combinatorial proof of one form of the MacWilliam identities for binary codes. (C) Elsevier Science Inc., 1997.
Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by C(f) = {Tr(af (x) + bx) x is an element of F-qm* :a, b is an eleme...
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Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by C(f) = {Tr(af (x) + bx) x is an element of F-qm* :a, b is an element of F-q(m)}, where q is a prime power, F-q(m) = F-q(m) \ punctured code, Tr is the trace function from F-q(m) to F-q, and f (x) is a function from F-q(m) to F-q(m) with f (0) = 0. Almost bent functions, quadratic functions and some monomials on F-2(m) were used in the first construction, and many families of binary linear codes with few weights were obtained in the literature. This paper studies some punctured codes of these binary codes. Several families of binary linear codes with few weights and new parameters are obtained in this paper. Several families of distance-optimal binary linear codes with new parameters are also produced in this paper.
A pragmatic Trellis coded MPSK on a Rayleigh fading channel is analyzed. This scheme allows bandwidth expansion ratio to be varied aiming at an optimization between complexity of the system design and improvement of c...
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A pragmatic Trellis coded MPSK on a Rayleigh fading channel is analyzed. This scheme allows bandwidth expansion ratio to be varied aiming at an optimization between complexity of the system design and improvement of coding gain. In order to vary the bandwidth expansion ratio, a punctured convolutional code is used. The performance of the proposed TC-2(m)PSK on a Rayleigh fading channel is theoretically analyzed. In the test examples, the BER performances of TC-QPSK and TC-8PSK are evaluated by theoretical analyses and computer simulations at the encoder parameters of K = 3 and r=3/4. The results show that the proposed scheme can attain better performance not only over the uncoded scheme but over the conventional Pragmatic TCM.
The puncturing and shortening techniques are two important approaches to constructing new linear codes from old ones. In the past 70 years, a lot of progress on the puncturing technique has been made, and many works o...
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The puncturing and shortening techniques are two important approaches to constructing new linear codes from old ones. In the past 70 years, a lot of progress on the puncturing technique has been made, and many works on punctured linear codes have been done. Many families of linear codes with interesting parameters have been obtained with the puncturing technique. However, little research on the shortening technique has been done and there are only a handful references on shortened linear codes. The first objective of this paper is to prove some general theory for shortened linear codes. The second objective is to study some shortened codes of the Hamming codes, Simplex codes, some Reed-Muller codes, and ovoid codes. Eleven families of optimal shortened codes over finite fields are presented in this paper. As a byproduct, five infinite families of 2-designs are also constructed from some of the shortened codes presented in this paper.
Subfield codes of linear codes over finite fields have recently attracted great attention due to their wide applications in secret sharing, authentication codes and association schemes. In this paper, we first present...
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Subfield codes of linear codes over finite fields have recently attracted great attention due to their wide applications in secret sharing, authentication codes and association schemes. In this paper, we first present a construction of 3-dimensional linear codes C-f over finite field F-2(m) parameterized by any Boolean function f. Then we determine explicitly the weight distributions of C-f, the punctured code (C) over tilde (f), as well as the corresponding subfield codes over F-2 for several classes of Boolean functions ff. In particular, we determine the weight distributions of subfield codes derived from r-plateaued functions. Moreover, the parameters of their dual codes are investigated, which contain length-optimal and dimension-optimal AMDS codes with respect to the sphere packing bound. We emphasize that the new codes are projective and contain binary self-complementary codes. As applications, some of the projective codes we present can be employed to construct s-sum sets for any odd integer s >1.
Locally recoverable codes (LRCs) were proposed for the recovery of data in distributed and cloud storage systems about nine years ago. A lot of progress on the study of LRCs has been made by now. However, there is a l...
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Locally recoverable codes (LRCs) were proposed for the recovery of data in distributed and cloud storage systems about nine years ago. A lot of progress on the study of LRCs has been made by now. However, there is a lack of general theory on the minimum locality of linear codes. In addition, the minimum locality of many known families of linear codes has not been studied in the literature. Motivated by these two facts, this paper develops some general theory about the minimum locality of linear codes, and investigates the minimum locality of a number of families of linear codes, such as q-ary Hamming codes, q-ary Simplex codes, generalized Reed-Muller codes, ovoid codes, maximum arc codes, the extended hyperoval codes, and near MDS codes. Many classes of both distance-optimal and dimension-optimal LRCs are presented in this paper. To this end, the concepts of linear locality and minimum linear locality are specified. The minimum linear locality of many families of linear codes are settled with the general theory developed in this paper.
Most codes with an algebraic decoding algorithm are derived from Reed-Solomon codes. They are obtained by taking equivalent codes, for example, generalized Reed-Solomon codes, or by using the so-called subfield subcod...
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Most codes with an algebraic decoding algorithm are derived from Reed-Solomon codes. They are obtained by taking equivalent codes, for example, generalized Reed-Solomon codes, or by using the so-called subfield subcode method, which leads to alternant codes over the underlying prime field, or over some intermediate subfield. The main advantage of these constructions is to preserve both the minimum distance and the decoding algorithm of the underlying Reed-Solomon code. In this paper, we explore in detail the subspace subcodes construction. This kind of codes was already studied in the particular case of cyclic Reed-Solomon codes. We extend this approach to any linear code over the extension of a finite field. We are interested in additive codes who are deeply connected to subfield subcodes. We characterize the duals of subspace subcodes. We introduce the notion of generalized subspace subcodes. We apply our results to generalized Reed-Solomon codes which leads to codes with interesting parameters, especially over a large alphabet. To conclude this paper, we discuss the security of the use of generalized subspace subcodes of Reed-Solomon codes in a cryptographic context.
We have investigated the feasibility of the 16QAM-OFDM transmission scheme by calculating its transmission performance by means of some evaluation models. These models have been proposed by standardization projects su...
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We have investigated the feasibility of the 16QAM-OFDM transmission scheme by calculating its transmission performance by means of some evaluation models. These models have been proposed by standardization projects such as ETSI-BRAN or ARIB-MMAC project. We evaluated the effects of the following four factors on transmission performance: (1) modulation scheme of each OFDM sub-carrier-channel;(2) FEC scheme. especially coding rate and coding scheme: (3) multipath fading environments;and (4) phase noise and nonlinear amplifier. By these evaluations, we have compiled a fundamental data in order to realize multimedia mobile access communication system based on 16QAM-OFDM transmission scheme.
Consider a complete flag {0} = C-0 2g + 2 rational points is isometry-dual if and only if the last code C-n in the flag is defined with functions of pole order at most n + 2g - 1. Using a different approach, we exten...
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Consider a complete flag punctured = C-0 < C-1 < center dot center dot center dot < C-n = F-n of one-point AG codes of length n over the finite field F. The codes are defined by evaluating functions with poles at a given point Q in points P-1, ... , P-n distinct from Q. A flag has the isometry-dual property if the given flag and the corresponding dual flag are the same up to isometry. For several curves, including the projective line, Hermitian curves, Suzuki curves, Ree curves, and the Klein curve over the field of eight elements, the maximal flag, obtained by evaluation in all rational points different from the point Q, is self-dual. More generally, we ask whether a flag obtained by evaluation in a proper subset of rational points is isometry-dual. In Geil et al. (2011) it is shown, for a curve of genus g, that a flag of one-point AG codes defined with a subset of n > 2g + 2 rational points is isometry-dual if and only if the last code C-n in the flag is defined with functions of pole order at most n + 2g - 1. Using a different approach, we extend this characterization to all subsets of size n >= 2g + 2. Moreover we show that this is best possible by giving examples of isometry-dual flags with n = 2g + 1 such that Cn is generated by functions of pole order at most n + 2g - 2. We also prove a necessary condition, formulated in terms of maximum sparse ideals of the Weierstrass semigroup of Q, under which a flag of punctured one-point AG codes inherits the isometry-dual property from the original unpunctured flag.
The weight spectra of high-rate punctured convolutional codes are evaluated under the hypothesis of a low-rate structure. This interpretation yields results slightly different from those obtained when weight spectra a...
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The weight spectra of high-rate punctured convolutional codes are evaluated under the hypothesis of a low-rate structure. This interpretation yields results slightly different from those obtained when weight spectra are evaluated assuming a true high-rate structure for punctured codes. The paper also extends the search for long memory punctured codes by providing new punctured codes of rates 4/5, 5/6, 6/7, and 7/8 with memories M ranging from 9 to 19.
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