Generalized Sphere-Packing Bound for Subblock-Constrained Codes

We apply the generalized sphere-packing bound to two classes of subblock-constrained codes. A la Fazeli et al. (2015), we made use of automorphism to significantly reduce the number of variables in the associated linear programming problem. In particular, we study binary constant subblock-composition codes (CSCCs), characterized by the property that the number of ones in each subblock is constant, and binary subblock energy-constrained codes (SECCs), characterized by the property that the number of ones in each subblock exceeds a certain threshold. For CSCCs, we show that the optimization problem is equivalent to finding the minimum of $N$ variables, where $N$ is independent of the number of subblocks. We then provide closed-form solutions for the generalized sphere-packing bounds for single- and double-error correcting CSCCs. For SECCs, we provide closed-form solutions for the generalized sphere-packing bounds for single errors in certain special cases. We also obtain improved bounds on the optimal asymptotic rate for CSCCs and SECCs, and provide numerical examples to highlight the improvement

Quantum Stabilizer Codes and Classical Linear Codes

We show that within any quantum stabilizer code there lurks a classical binary linear code with similar error-correcting capabilities, thereby demonstrating new connections between quantum codes and classical codes. Using this result -- which applies to degenerate as well as nondegenerate codes -- previously established necessary conditions for classical linear codes can be easily translated into necessary conditions for quantum stabilizer codes. Examples of specific consequences are: for a quantum channel subject to a delta-fraction of errors, the best asymptotic capacity attainable by any stabilizer code cannot exceed H(1/2 + sqrt(2*delta*(1-2*delta))); and, for the depolarizing channel with fidelity parameter delta, the best asymptotic capacity attainable by any stabilizer code cannot exceed 1-H(delta).Comment: 17 pages, ReVTeX, with two figure

Long Nonbinary Codes Exceeding the Gilbert - Varshamov Bound for any Fixed Distance

Let A(q,n,d) denote the maximum size of a q-ary code of length n and distance d. We study the minimum asymptotic redundancy \rho(q,n,d)=n-log_q A(q,n,d) as n grows while q and d are fixed. For any d and q<=d-1, long algebraic codes are designed that improve on the BCH codes and have the lowest asymptotic redundancy \rho(q,n,d) <= ((d-3)+1/(d-2)) log_q n known to date. Prior to this work, codes of fixed distance that asymptotically surpass BCH codes and the Gilbert-Varshamov bound were designed only for distances 4,5 and 6.Comment: Submitted to IEEE Trans. on Info. Theor

Principles and Parameters: a coding theory perspective

We propose an approach to Longobardi's parametric comparison method (PCM) via the theory of error-correcting codes. One associates to a collection of languages to be analyzed with the PCM a binary (or ternary) code with one code words for each language in the family and each word consisting of the binary values of the syntactic parameters of the language, with the ternary case allowing for an additional parameter state that takes into account phenomena of entailment of parameters. The code parameters of the resulting code can be compared with some classical bounds in coding theory: the asymptotic bound, the Gilbert-Varshamov bound, etc. The position of the code parameters with respect to some of these bounds provides quantitative information on the variability of syntactic parameters within and across historical-linguistic families. While computations carried out for languages belonging to the same family yield codes below the GV curve, comparisons across different historical families can give examples of isolated codes lying above the asymptotic bound.Comment: 11 pages, LaTe

Non-asymptotic Upper Bounds for Deletion Correcting Codes

Explicit non-asymptotic upper bounds on the sizes of multiple-deletion correcting codes are presented. In particular, the largest single-deletion correcting code for $q$-ary alphabet and string length $n$ is shown to be of size at most $\frac{q^n-q}{(q-1)(n-1)}$. An improved bound on the asymptotic rate function is obtained as a corollary. Upper bounds are also derived on sizes of codes for a constrained source that does not necessarily comprise of all strings of a particular length, and this idea is demonstrated by application to sets of run-length limited strings. The problem of finding the largest deletion correcting code is modeled as a matching problem on a hypergraph. This problem is formulated as an integer linear program. The upper bound is obtained by the construction of a feasible point for the dual of the linear programming relaxation of this integer linear program. The non-asymptotic bounds derived imply the known asymptotic bounds of Levenshtein and Tenengolts and improve on known non-asymptotic bounds. Numerical results support the conjecture that in the binary case, the Varshamov-Tenengolts codes are the largest single-deletion correcting codes.Comment: 18 pages, 4 figure