Enhanced Recursive Reed-Muller Erasure Decoding

Recent work have shown that Reed-Muller (RM) codes achieve the erasure channel capacity. However, this performance is obtained with maximum-likelihood decoding which can be costly for practical applications. In this paper, we propose an encoding/decoding scheme for Reed-Muller codes on the packet erasure channel based on Plotkin construction. We present several improvements over the generic decoding. They allow, for a light cost, to compete with maximum-likelihood decoding performance, especially on high-rate codes, while significantly outperforming it in terms of speed

Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms

The local minimum degree of a graph is the minimum degree that can be reached by means of local complementation. For any n, there exist graphs of order n which have a local minimum degree at least 0.189n, or at least 0.110n when restricted to bipartite graphs. Regarding the upper bound, we show that for any graph of order n, its local minimum degree is at most 3n/8+o(n) and n/4+o(n) for bipartite graphs, improving the known n/2 upper bound. We also prove that the local minimum degree is smaller than half of the vertex cover number (up to a logarithmic term). The local minimum degree problem is NP-Complete and hard to approximate. We show that this problem, even when restricted to bipartite graphs, is in W[2] and FPT-equivalent to the EvenSet problem, which W[1]-hardness is a long standing open question. Finally, we show that the local minimum degree is computed by a O*(1.938^n)-algorithm, and a O*(1.466^n)-algorithm for the bipartite graphs

Linear Boolean classification, coding and "the critical problem"

The problem of constructing a minimal rank matrix over GF(2) whose kernel does not intersect a given set S is considered. In the case where S is a Hamming ball centered at 0, this is equivalent to finding linear codes of largest dimension. For a general set, this is an instance of "the critical problem" posed by Crapo and Rota in 1970. This work focuses on the case where S is an annulus. As opposed to balls, it is shown that an optimal kernel is composed not only of dense but also of sparse vectors, and the optimal mixture is identified in various cases. These findings corroborate a proposed conjecture that for annulus of inner and outer radius nq and np respectively, the optimal relative rank is given by (1-q)H(p/(1-q)), an extension of the Gilbert-Varshamov bound H(p) conjectured for Hamming balls of radius np

Single query learning from abelian and non-abelian Hamming distance oracles

We study the problem of identifying an n-bit string using a single quantum query to an oracle that computes the Hamming distance between the query and hidden strings. The standard action of the oracle on a response register of dimension r is by powers of the cycle (1...r), all of which, of course, commute. We introduce a new model for the action of an oracle--by general permutations in S_r--and explore how the success probability depends on r and on the map from Hamming distances to permutations. In particular, we prove that when r = 2, for even n the success probability is 1 with the right choice of the map, while for odd n the success probability cannot be 1 for any choice. Furthermore, for small odd n and r = 3, we demonstrate numerically that the image of the optimal map generates a non-abelian group of permutations.Comment: 14 page

Small codes

In 1930, Tammes posed the problem of determining $\rho(r, n)$, the minimum over all sets of $n$ unit vectors in $\mathbb{R}^r$ of their maximum pairwise inner product. In 1955, Rankin determined $\rho(r,n)$ whenever $n \leq 2r$ and in this paper we show that $\rho(r, 2r + k ) \geq \frac{\left(\frac{8}{27}k + 1\right)^{1/3} - 1}{2r + k}$, answering a question of Bukh and Cox. As a consequence, we conclude that the maximum size of a binary code with block length $r$ and minimum Hamming distance $(r-j)/2$ is at most $(2 + o(1))r$ when $j = o(r^{1/3})$, resolving a conjecture of Tiet\"av\"ainen from 1980 in a strong form. Furthermore, using a recently discovered connection to binary codes, this yields an analogous result for set-coloring Ramsey numbers of triangles.Comment: 6 page