In this paper, we examine the locality property of the original Fractional Repetition(FR) codes and propose two constructions for FR codes with better locality. For this, we first derive the capacity of the FR codes with locality 2, that is the maximum size of the file that can be stored. Construction 1 generates an FR code with repetition degree 2 and locality 2. This code is optimal in the se...

Product codes based on binary block codes are considered. The aim is to use simple block codes as component codes in the constructions. Then, by using interleaving, large codes can be obtained. The fundamental aim of the interleaving is to obtain codes with a structure that is suitable for iterative decoding. We present interleaving strategies for iterative (turbo) decoding of block codes. A si...

A convolutional code has memory over a short block length. This memory results in encoded output symbols that depend not only on the present input, but also on past inputs. An (n,k,m) convolutional code is implemented using k-input, n-output linear sequential system with a shift-register having m stages. In practice k and n are small and m is large to achieve low error probabilities. In the par...

Permutation codes are vector quantizers whose codewords are related by permutations and, in one variant, sign changes. Asymptotically, as the vector dimension grows, optimal Variant I permutation code design is identical to optimal entropy-constrained scalar quantizer (ECSQ) design. However, contradicting intuition and previously published assertions, there are finite block length permutation c...

For example, CH the family of Hamming code is a family of codes with ni = 2 − 1, ki = 2 − i − 1, di = 3 and R(CH) = 1, δ(CH) = 0. We will mostly work with family of codes from now on. This is necessary as we will study the asymptotic behavior of algorithms for codes, which does not make sense for a fixed code. For example, when we say we say that a decoding algorithm for a code C takes O(n) tim...

An open question about Gabidulin codes is whether polynomial-time list decoding beyond half the minimum distance is possible or not. In this contribution, we give a lower and an upper bound on the list size, i.e., the number of codewords in a ball around the received word. The lower bound shows that if the radius of this ball is greater than the Johnson radius, this list size can be exponential...

Let K = Fq be a finite field. We introduce a family of projective Reed-Mullertype codes called projective Segre codes. Then we study their basic parameters and show that they are direct products of projective Reed-Muller-type codes. It turns out that the direct product of two projective Reed-Muller-type codes is again a projective Reed-Muller-type code. As a consequence we recover some results ...

Consider a finite set of q elements and containing a distinguished element ”zero”. The choice of a set does not matter in our context and we will use the set Zq of integers modulo q. Let Z q be the set of n-tuples (or vectors) over Zq and Z n,w q be the set of n-tuples over Zq of Hamming weight w. A code is called constant-weight if all the codewords have the same weight w. A code is called equ...

A method is presented for obtaining the error probability for block codes. The method is based on the eigenvalueeigenvector properties of the code correlation matrix. It is found that under a unary transformation and for an additive white Gaussian noise environment, the performance evaluation of a block code becomes a one-dimensional problem in which only one eigenvalue and its corresponding ei...

In Codeplay’s Sieve C++, the programmer can place code inside a “sieve block” thereby instructing the compiler to delay writes to global memory and apply them in order on exit from the block. The semantics of sieve blocks makes code more amenable to automatic parallelisation. However, strictly queueing writes until the end of a sieve block incurs overheads and is typically unnecessary. If the p...