Linear Code

Abstract. In this paper we prove that a set of points (in a projective space over a finite field of q elements), which is incident with 0 mod r points of every hyperplane, has at least (r−1)q+(p−1)r points, where 1 < r < q = ph, p prime. An immediate corollary of this theorem is that a linear code whose weights and length have a common divisor r < q and whose dual minimum distance is at least 3, has length at least (r − 1)q + (p − 1)r. The theorem, which is sharp in some cases, is a strong generalisation of an earlier result on the non-existence of maximal arcs in projective planes; the proof involves polynomials over finite fields, and is a streamlined and more transparent version of the earlier one. 1.

Abstract-In this paper we examine several systematic (16,s) codes capable of double error correction (DEC) and triple adjacent error detection. In particular, we examine a linear version of the Nordstrom-Robinson (NR) code in the light of new findings by Hammons and others, and ...

The paper presents the results of numerical analyses carried out with FLAC finite difference code aiming at investigating the seismic response of rockfill dams. In particular the hysteretic damping model, recently incorporated within the code, coupled with a perfectly plastic yield criterion, was employed. As first step, 1D and 2D calibration analyses were performed and comparisons with the results supplied by well known linear equivalent and fully non linear codes were carried out. Then the seismic response of E1 Infiernillo rockfill dam was investigated during two weak and strong seismic events. Benefits and shortcomings of using the hysteretic damping model are discussed in the light of the results obtained from calibration studies and field-scale analyses.

This short article gives an introduction to binary linear block codes for error detection and error correction.  Various codes are discussed and compared, including repetition codes, Hamming codes, and Reed-Muller codes.

A version of this article was published by the Radio Society of Great Britain in their members' magazine/journal, 'RadCom'.

In this paper, we construct four binary linear codes closely connected with certain exponential sums over the finite field F_q and F_q-{0,1}. Here q is a power of two. Then we obtain four recursive formulas for the power moments of Kloosterman sums in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of the exponential sums obtained earlier.

In an earlier paper the authors studied simplex codes of type α and β over ${\mathbb{Z}}_4$ and obtained some known binary linear and nonlinear codes as Gray images of these codes. In this correspondence, we study weight distributions of simplex codes of type α and β over ${\mathbb{Z}}_{{2^s}}.$ The generalized Gray map is then used to construct binary codes. The linear codes meet the Griesmer bound and a few non-linear codes are obtained that meet the Plotkin/Johnson bound. We also give the weight hierarchies of the first order Reed-Muller codes over ${\mathbb{Z}}_{2^{s}}.$ The above codes are also shown to satisfy the chain condition.

New quaternary Plotkin constructions are given and are used to obtain new families of quaternary codes. The parameters of the obtained codes, such as the length, the dimension and the minimum distance are studied. Using these constructions new families of quaternary Reed-Muller codes are built with the peculiarity that after using the Gray map the obtained Z4-linear codes have the

Low-density parity-check (LDPC) codes in their broader-sense definition are linear codes whose parity-check matrices have fewer 1s than 0s. Finding their minimum distance is therefore in general an NP-hard problem. We propose a randomized algorithm called nearest nonzero codeword search (NNCS) approach to tackle this problem for iteratively decodable LDPC codes. The principle of the NNCS approach is to search codewords locally around the all-zero codeword perturbed by minimal noise, anticipating that the resultant nearest nonzero codewords will most likely contain the minimum-Hamming- weight codeword whose Hamming weight is equal to the minimum distance of the linear code. This approach has its roots in Berrou et al.'s error-impulse method and a form of Fossorier's list decoding for LDPC codes.

In this paper, we introduce the concept of dual universality of hash functions and present its applications to quantum cryptography. We begin by establishing the one-to-one correspondence between a linear function family {\cal F} and a code family {\cal C}, and thereby defining \varepsilon-almost dual universal_2 hash functions, as a generalization of the conventional universal_2 hash functions. Then we show that this generalized (and thus broader) class of hash functions is in fact sufficient for the security of quantum cryptography. This result can be explained in two different formalisms. First, by noting its relation to the \delta-biased family introduced by Dodis and Smith, we demonstrate that Renner's two-universal hashing lemma is generalized to our class of hash functions. Next, we prove that the proof technique by Shor and Preskill can be applied to quantum key distribution (QKD) systems that use our generalized class of hash functions for privacy amplification. While Shor-Preskill formalism requires an implementer of a QKD system to explicitly construct a linear code of the Calderbank-Shor-Steane type, this result removes the existing difficulty of the construction a linear code of CSS code by replacing it by the combination of an ordinary classical error correcting code and our proposed hash function. We also show that a similar result applies to the quantum wire-tap channel. Finally we compare our results in the two formalisms and show that, in typical QKD scenarios, the Shor-Preskill--type argument gives better security bounds in terms of the trace distance and Holevo information, than the method based on the \delta-biased family.