On the Minimum Distance of Array-Based Spatially-Coupled Low-Density Parity-Check Codes

An array low-density parity-check (LDPC) code is a quasi-cyclic LDPC code specified by two integers $q$ and $m$, where $q$ is an odd prime and $m \leq q$. The exact minimum distance, for small $q$ and $m$, has been calculated, and tight upper bounds on it for $m \leq 7$ have been derived. In this work, we study the minimum distance of the spatially-coupled version of these codes. In particular, several tight upper bounds on the optimal minimum distance for coupling length at least two and $m=3,4,5$, that are independent of $q$ and that are valid for all values of $q \geq q_0$ where $q_0$ depends on $m$, are presented. Furthermore, we show by exhaustive search that by carefully selecting the edge spreading or unwrapping procedure, the minimum distance (when $q$ is not very large) can be significantly increased, especially for $m=5$.Comment: 5 pages. To be presented at the 2015 IEEE International Symposium on Information Theory, June 14-19, 2015, Hong Kon

Further Results on Quadratic Permutation Polynomial-Based Interleavers for Turbo Codes

An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions are of particular interest because they admit analytical designs and simple, practical hardware implementation. Also, the recently proposed quadratic permutation polynomial (QPP) based interleavers by Sun and Takeshita (IEEE Trans. Inf. Theory, Jan. 2005) provide excellent performance for short-to-medium block lengths, and have been selected for the 3GPP LTE standard. In this work, we derive some upper bounds on the best achievable minimum distance dmin of QPP-based conventional binary turbo codes (with tailbiting termination, or dual termination when the interleaver length N is sufficiently large) that are tight for larger block sizes. In particular, we show that the minimum distance is at most 2(2^{\nu +1}+9), independent of the interleaver length, when the QPP has a QPP inverse, where {\nu} is the degree of the primitive feedback and monic feedforward polynomials. However, allowing the QPP to have a larger degree inverse may give strictly larger minimum distances (and lower multiplicities). In particular, we provide several QPPs with an inverse degree of at least three for some of the 3GPP LTE interleaver lengths giving a dmin with the 3GPP LTE constituent encoders which is strictly larger than 50. For instance, we have found a QPP for N=6016 which gives an estimated dmin of 57. Furthermore, we provide the exact minimum distance and the corresponding multiplicity for all 3GPP LTE turbo codes (with dual termination) which shows that the best minimum distance is 51. Finally, we compute the best achievable minimum distance with QPP interleavers for all 3GPP LTE interleaver lengths N <= 4096, and compare the minimum distance with the one we get when using the 3GPP LTE polynomials.Comment: Submitted to IEEE Trans. Inf. Theor

Lengthening and Extending Binary Private Information Retrieval Codes

It was recently shown by Fazeli et al. that the storage overhead of a traditional $t$-server private information retrieval (PIR) protocol can be significantly reduced using the concept of a $t$-server PIR code. In this work, we show that a family of $t$-server PIR codes (with increasing dimensions and blocklengths) can be constructed from an existing $t$-server PIR code through lengthening by a single information symbol and code extension by at most $\bigl\lceil t/2\bigr\rceil$ code symbols. Furthermore, by extending a code construction notion from Steiner systems by Fazeli et al., we obtain a specific family of $t$-server PIR codes. Based on a code construction technique that lengthens and extends a $t$-server PIR code simultaneously, a basic algorithm to find good (i.e., small blocklength) $t$-server PIR codes is proposed. For the special case of $t=5$, we find provably optimal PIR codes for code dimensions $k\leq 6$, while for all $7\leq k\leq 32$ we find codes of smaller blocklength than the best known codes from the literature. Furthermore, in the case of $t = 8$, we also find better codes for $k = 5, 6, 11, 12$. Numerical results show that most of the best found $5$-server PIR codes can be constructed from the proposed family of codes connected to Steiner systems.Comment: The shorter version of this paper will appear in the proceedings of 2018 International Zurich Seminar on Information and Communicatio

Analysis of Spatially-Coupled Counter Braids

A counter braid (CB) is a novel counter architecture introduced by Lu et al. in 2007 for per-flow measurements on high-speed links. CBs achieve an asymptotic compression rate (under optimal decoding) that matches the entropy lower bound of the flow size distribution. Spatially-coupled CBs (SC-CBs) have recently been proposed. In this work, we further analyze single-layer CBs and SC-CBs using an equivalent bipartite graph representation of CBs. On this equivalent representation, we show that the potential and area thresholds are equal. We also show that the area under the extended belief propagation (BP) extrinsic information transfer curve (defined for the equivalent graph), computed for the expected residual CB graph when a peeling decoder equivalent to the BP decoder stops, is equal to zero precisely at the area threshold. This, combined with simulations and an asymptotic analysis of the Maxwell decoder, leads to the conjecture that the area threshold is in fact equal to the Maxwell decoding threshold and hence a lower bound on the maximum a posteriori (MAP) decoding threshold. Finally, we present some numerical results and give some insight into the apparent gap of the BP decoding threshold of SC-CBs to the conjectured lower bound on the MAP decoding threshold.Comment: To appear in the IEEE Information Theory Workshop, Jeju Island, Korea, October 201

On the Minimum/Stopping Distance of Array Low-Density Parity-Check Codes

In this work, we study the minimum/stopping distance of array low-density parity-check (LDPC) codes. An array LDPC code is a quasi-cyclic LDPC code specified by two integers q and m, where q is an odd prime and m <= q. In the literature, the minimum/stopping distance of these codes (denoted by d(q,m) and h(q,m), respectively) has been thoroughly studied for m <= 5. Both exact results, for small values of q and m, and general (i.e., independent of q) bounds have been established. For m=6, the best known minimum distance upper bound, derived by Mittelholzer (IEEE Int. Symp. Inf. Theory, Jun./Jul. 2002), is d(q,6) <= 32. In this work, we derive an improved upper bound of d(q,6) <= 20 and a new upper bound d(q,7) <= 24 by using the concept of a template support matrix of a codeword/stopping set. The bounds are tight with high probability in the sense that we have not been able to find codewords of strictly lower weight for several values of q using a minimum distance probabilistic algorithm. Finally, we provide new specific minimum/stopping distance results for m <= 7 and low-to-moderate values of q <= 79.Comment: To appear in IEEE Trans. Inf. Theory. The material in this paper was presented in part at the 2014 IEEE International Symposium on Information Theory, Honolulu, HI, June/July 201