Multiset Combinatorial Batch Codes

Batch codes, first introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai, mimic a distributed storage of a set of $n$ data items on $m$ servers, in such a way that any batch of $k$ data items can be retrieved by reading at most some $t$ symbols from each server. Combinatorial batch codes, are replication-based batch codes in which each server stores a subset of the data items. In this paper, we propose a generalization of combinatorial batch codes, called multiset combinatorial batch codes (MCBC), in which $n$ data items are stored in $m$ servers, such that any multiset request of $k$ items, where any item is requested at most $r$ times, can be retrieved by reading at most $t$ items from each server. The setup of this new family of codes is motivated by recent work on codes which enable high availability and parallel reads in distributed storage systems. The main problem under this paradigm is to minimize the number of items stored in the servers, given the values of $n,m,k,r,t$, which is denoted by $N(n,k,m,t;r)$. We first give a necessary and sufficient condition for the existence of MCBCs. Then, we present several bounds on $N(n,k,m,t;r)$ and constructions of MCBCs. In particular, we determine the value of $N(n,k,m,1;r)$ for any $n\geq \left\lfloor\frac{k-1}{r}\right\rfloor{m\choose k-1}-(m-k+1)A(m,4,k-2)$, where $A(m,4,k-2)$ is the maximum size of a binary constant weight code of length $m$, distance four and weight $k-2$. We also determine the exact value of $N(n,k,m,1;r)$ when $r\in\{k,k-1\}$ or $k=m$

New Constant-Weight Codes from Propagation Rules

This paper proposes some simple propagation rules which give rise to new binary constant-weight codes.Comment: 4 page

Multiply Constant-Weight Codes and the Reliability of Loop Physically Unclonable Functions

We introduce the class of multiply constant-weight codes to improve the reliability of certain physically unclonable function (PUF) response. We extend classical coding methods to construct multiply constant-weight codes from known $q$-ary and constant-weight codes. Analogues of Johnson bounds are derived and are shown to be asymptotically tight to a constant factor under certain conditions. We also examine the rates of the multiply constant-weight codes and interestingly, demonstrate that these rates are the same as those of constant-weight codes of suitable parameters. Asymptotic analysis of our code constructions is provided

Group Divisible Codes and Their Application in the Construction of Optimal Constant-Composition Codes of Weight Three

The concept of group divisible codes, a generalization of group divisible designs with constant block size, is introduced in this paper. This new class of codes is shown to be useful in recursive constructions for constant-weight and constant-composition codes. Large classes of group divisible codes are constructed which enabled the determination of the sizes of optimal constant-composition codes of weight three (and specified distance), leaving only four cases undetermined. Previously, the sizes of constant-composition codes of weight three were known only for those of sufficiently large length.Comment: 13 pages, 1 figure, 4 table

Importance of Symbol Equity in Coded Modulation for Power Line Communications

The use of multiple frequency shift keying modulation with permutation codes addresses the problem of permanent narrowband noise disturbance in a power line communications system. In this paper, we extend this coded modulation scheme based on permutation codes to general codes and introduce an additional new parameter that more precisely captures a code's performance against permanent narrowband noise. As a result, we define a new class of codes, namely, equitable symbol weight codes, which are optimal with respect to this measure