Projective codes meeting the Griesmer bound

AbstractWe present a brief survey of projective codes meeting the Griesmer bound. Methods for constructing large families of codes as well as sporadic codes meeting the bound are given. Current research on the classification of codes meeting the Griesmer bound is also presented

Update-Efficiency and Local Repairability Limits for Capacity Approaching Codes

Motivated by distributed storage applications, we investigate the degree to which capacity achieving encodings can be efficiently updated when a single information bit changes, and the degree to which such encodings can be efficiently (i.e., locally) repaired when single encoded bit is lost. Specifically, we first develop conditions under which optimum error-correction and update-efficiency are possible, and establish that the number of encoded bits that must change in response to a change in a single information bit must scale logarithmically in the block-length of the code if we are to achieve any nontrivial rate with vanishing probability of error over the binary erasure or binary symmetric channels. Moreover, we show there exist capacity-achieving codes with this scaling. With respect to local repairability, we develop tight upper and lower bounds on the number of remaining encoded bits that are needed to recover a single lost bit of the encoding. In particular, we show that if the code-rate is $\epsilon$ less than the capacity, then for optimal codes, the maximum number of codeword symbols required to recover one lost symbol must scale as $\log1/\epsilon$. Several variations on---and extensions of---these results are also developed.Comment: Accepted to appear in JSA

Entanglement-Assisted Quantum Communication Beating the Quantum Singleton Bound

Brun, Devetak, and Hsieh [Science 314, 436 (2006)] demonstrated that pre-shared entanglement between sender and receiver enables quantum communication protocols that have better parameters than schemes without the assistance of entanglement. Subsequently, the same authors derived a version of the so-called quantum Singleton bound that relates the parameters of the entanglement-assisted quantum-error correcting codes proposed by them. We present a new entanglement-assisted quantum communication scheme with parameters violating this bound in certain ranges

The weight enumerator polynomials of the lifted codes of the projective Solomon-Stiffler codes

Determining the weight distribution of a code is an old and fundamental topic in coding theory that has been thoroughly studied. In 1977, Helleseth, Kl{\o}ve, and Mykkeltveit presented a weight enumerator polynomial of the lifted code over $\mathbb{F}_{q^\ell}$ of a $q$-ary linear code with significant combinatorial properties, which can determine the support weight distribution of this linear code. The Solomon-Stiffler codes are a family of famous Griesmer codes, which were proposed by Solomon and Stiffler in 1965. In this paper, we determine the weight enumerator polynomials of the lifted codes of the projective Solomon-Stiffler codes using some combinatorial properties of subspaces. As a result, we determine the support weight distributions of the projective Solomon-Stiffler codes. In particular, we determine the weight hierarchies of the projective Solomon-Stiffler codes.Comment: This manuscript was first submitted on September 9, 202

Ternary linear codes

iii+69hlm.;24c

Covering Radius 1985-1994

We survey important developments in the theory of covering radius during the period 1985-1994. We present lower bounds, constructions and upper bounds, the linear and nonlinear cases, density and asymptotic results, normality, specific classes of codes, covering radius and dual distance, tables, and open problems

Publications of the Jet Propulsion Laboratory, July 1964 through June 1965

JPL publications bibliography with abstracts - reports on DSIF, Mariner program, Ranger project, Surveyor project, and other space programs, and space science