Error Correction for Index Coding With Coded Side Information

Index coding is a source coding problem in which a broadcaster seeks to meet the different demands of several users, each of whom is assumed to have some prior information on the data held by the sender. If the sender knows its clients' requests and their side-information sets, then the number of packet transmissions required to satisfy all users' demands can be greatly reduced if the data is encoded before sending. The collection of side-information indices as well as the indices of the requested data is described as an instance of the index coding with side-information (ICSI) problem. The encoding function is called the index code of the instance, and the number of transmissions employed by the code is referred to as its length. The main ICSI problem is to determine the optimal length of an index code for and instance. As this number is hard to compute, bounds approximating it are sought, as are algorithms to compute efficient index codes. Two interesting generalizations of the problem that have appeared in the literature are the subject of this work. The first of these is the case of index coding with coded side information, in which linear combinations of the source data are both requested by and held as users' side-information. The second is the introduction of error-correction in the problem, in which the broadcast channel is subject to noise. In this paper we characterize the optimal length of a scalar or vector linear index code with coded side information (ICCSI) over a finite field in terms of a generalized min-rank and give bounds on this number based on constructions of random codes for an arbitrary instance. We furthermore consider the length of an optimal error correcting code for an instance of the ICCSI problem and obtain bounds on this number, both for the Hamming metric and for rank-metric errors. We describe decoding algorithms for both categories of errors

A Reed-Solomon Coded DS-CDMA System Using Noncoherent M-ary Orthogonal Modulation over Multipath Fading Channels

The performance of Reed–Solomon (RS) coded direct-sequence code division multiple-access (DS-CDMA) systems using noncoherent M-ary orthogonal modulation is investigated over multipath Rayleigh fading channels. Diversity reception techniques with equal gain combining (EGC) or selection combining (SC) are invoked and the related performance is evaluated for both uncoded and coded DS-CDMA systems. "Errors-and-erasures" decoding is considered, where the erasures are based on Viterbi’s so-called ratio threshold test (RTT). The probability density functions (PDF) of the ratio associated with the RTT conditioned on both the correct detection and erroneous detection of the M-ary signals are derived. These PDFs are then used for computing the codeword decoding error probability of the RS coded DS-CDMA system using "errors-and-erasures" decoding. Furthermore, the performance of the "errors-and-erasures" decoding technique employing the RTT is compared to that of "error-correction-only" decoding refraining from using side-information over multipath Rayleigh fading channels. As expected, the numerical results show that when using "errors-and-erasures" decoding, RS codes of a given code rate can achieve a higher coding gain than without erasure information. Index Terms—Direct sequence code division multiple-access, error-correction-only decoding, errors-and-erasures decoding, noncoherent $M$-ary orthogonal signaling, ratio threshold test, Reed–Solomon codes

On Boolean functions, symmetric cryptography and algebraic coding theory

In the first part of this thesis we report results about some “linear” trapdoors that can be embedded in a block cipher. In particular we are interested in any block cipher which has invertible S-boxes and that acts as a permutation on the message space, once the key is chosen. The message space is a vector space and we can endow it with alternative operations (hidden sums) for which the structure of vector space is preserved. Each of this operation is related to a different copy of the affine group. So, our block cipher could be affine with respect to one of these hidden sums. We show conditions on the S-box able to prevent a type of trapdoors based on hidden sums, in particular we introduce the notion of Anti-Crooked function. Moreover we shows some properties of the translation groups related to these hidden sums, characterizing those that are generated by affine permutations. In that case we prove that hidden sum trapdoors are practical and we can perform a global reconstruction attack. We also analyze the role of the mixing layer obtaining results suggesting the possibility to have undetectable hidden sum trapdoors using MDS mixing layers. In the second part we take into account the index coding with side information (ICSI) problem. Firstly we investigate the optimal length of a linear index code, that is equal to the min-rank of the hypergraph related to the instance of the ICSI problem. In particular we extend the the so-called Sandwich Property from graphs to hypergraphs and also we give an upper bound on the min-rank of an hypergraph taking advantage of incidence structures such as 2-designs and projective planes. Then we consider the more general case when the side information are coded, the index coding with coded side information (ICCSI) problem. We extend some results on the error correction index codes to the ICCSI problem case and a syndrome decoding algorithm is also given