On the high density behavior of Hamming codes with fixed minimum distance

We discuss the high density behavior of a system of hard spheres of diameter d on the hypercubic lattice of dimension n, in the limit n -> oo, d -> oo, d/n=delta. The problem is relevant for coding theory. We find a solution to the equations describing the liquid up to very large values of the density, but we show that this solution gives a negative entropy for the liquid phase when the density is large enough. We then conjecture that a phase transition towards a different phase might take place, and we discuss possible scenarios for this transition. Finally we discuss the relation between our results and known rigorous bounds on the maximal density of the system.Comment: 15 pages, 6 figure

A Construction of Quantum LDPC Codes from Cayley Graphs

We study a construction of Quantum LDPC codes proposed by MacKay, Mitchison and Shokrollahi. It is based on the Cayley graph of Fn together with a set of generators regarded as the columns of the parity-check matrix of a classical code. We give a general lower bound on the minimum distance of the Quantum code in $\mathcal{O}(dn^2)$ where d is the minimum distance of the classical code. When the classical code is the $[n, 1, n]$ repetition code, we are able to compute the exact parameters of the associated Quantum code which are $[[2^n, 2^{\frac{n+1}{2}}, 2^{\frac{n-1}{2}}]]$.Comment: The material in this paper was presented in part at ISIT 2011. This article is published in IEEE Transactions on Information Theory. We point out that the second step of the proof of Proposition VI.2 in the published version (Proposition 25 in the present version and Proposition 18 in the ISIT extended abstract) is not strictly correct. This issue is addressed in the present versio

Upper bound on list-decoding radius of binary codes

Consider the problem of packing Hamming balls of a given relative radius subject to the constraint that they cover any point of the ambient Hamming space with multiplicity at most $L$. For odd $L\ge 3$ an asymptotic upper bound on the rate of any such packing is proven. Resulting bound improves the best known bound (due to Blinovsky'1986) for rates below a certain threshold. Method is a superposition of the linear-programming idea of Ashikhmin, Barg and Litsyn (that was used previously to improve the estimates of Blinovsky for $L=2$) and a Ramsey-theoretic technique of Blinovsky. As an application it is shown that for all odd $L$ the slope of the rate-radius tradeoff is zero at zero rate.Comment: IEEE Trans. Inform. Theory, accepte

Error Patterns

In coding theory the problem of decoding focuses on error vectors. In the simplest situation code words are $(0,1)$-vectors, as are the received messages and the error vectors. Comparison of a received word with the code words yields a set of error vectors. In deciding on the original code word, usually the one for which the error vector has minimum Hamming weight is chosen. In this note some remarks are made on the problem of the elements 1 in the error vector, that may enable unique decoding, in case two or more code words have the same Hamming distance to the received message word, thus turning error detection into error correction. The essentially new aspect is that code words, message words and error vectors are put in one-one correspondence with graphs

Cavity approach to sphere packing in Hamming space

In this paper we study the hard sphere packing problem in the Hamming space by the cavity method. We show that both the replica symmetric and the replica symmetry breaking approximations give maximum rates of packing that are asymptotically the same as the lower bound of Gilbert and Varshamov. Consistently with known numerical results, the replica symmetric equations also suggest a crystalline solution, where for even diameters the spheres are more likely to be found in one of the subspaces (even or odd) of the Hamming space. These crystalline packings can be generated by a recursive algorithm which finds maximum packings in an ultra-metric space. Finally, we design a message passing algorithm based on the cavity equations to find dense packings of hard spheres. Known maximum packings are reproduced efficiently in non trivial ranges of dimensions and number of spheres.Comment: 23 pages, 11 figures; typos correcte