Algebraic codes that achieve list decoding capacity were recently constructed by a careful folding'' of the Reed-Solomon code. The low-degree'' nature of this folding operation was crucial to the list decoding algorithm. We show how such folding schemes conducive to list decoding arise out of the Artin-Frobenius automorphism at primes in Galois extensions. Using this approach, we construct new folded algebraic-geometric codes for list decoding based on cyclotomic function fields with a cyclic Galois group. Such function fields are obtained by adjoining torsion points of the Carlitz action of an irreducible @math . The Reed-Solomon case corresponds to the simplest such extension (corresponding to the case @math ). In the general case, we need to descend to the fixed field of a suitable Galois subgroup in order to ensure the existence of many degree one places that can be used for encoding. Our methods shed new light on algebraic codes and their list decoding, and lead to new codes achieving list decoding capacity. Quantitatively, these codes provide list decoding (and list recovery soft decoding) guarantees similar to folded Reed-Solomon codes but with an alphabet size that is only polylogarithmic in the block length. In comparison, for folded RS codes, the alphabet size is a large polynomial in the block length. This has applications to fully explicit (with no brute-force search) binary concatenated codes for list decoding up to the Zyablov radius.
Independent of our work, Huang and Narayanan @cite_20 also consider AG codes constructed from Galois extensions, and observe how automorphisms of large order can be used for folding such codes. To our knowledge, the only instantiation of this approach that improves on folded RS codes is the one based on cyclotomic function fields from our work. As an alternate approach, they also propose a decoding method that works with folding via automorphisms of small order. This involves computing several coefficients of the power series expansion of the message function at a low-degree place. Unfortunately, piecing together these coefficients into a function could lead to an exponential list size bound. The authors suggest a heuristic assumption under which they can show that for a random received word, the expected list size and running time are polynomially bounded.
@cite_20
1637999745 2130507035 1909826856 2599155613
We describe a new class of list decodable codes based on Galois extensions of function fields and present a list decoding algorithm. These codes are obtained as a result of folding the set of rational places of a function field using certain elements (automorphisms) from the Galois group of the extension. This work is an extension of Folded Reed Solomon codes to the setting of Algebraic Geometric codes. We describe two constructions based on this framework depending on if the order of the automorphism used to fold the code is large or small compared to the block length. When the automorphism is of large order, the codes have polynomially bounded list size in the worst case. This construction gives codes of rate @math over an alphabet of size independent of block length that can correct a fraction of @math errors subject to the existence of asymptotically good towers of function fields with large automorphisms. The second construction addresses the case when the order of the element used to fold is small compared to the block length. In this case a heuristic analysis shows that for a random received word, the expected list size and the running time of the decoding algorithm are bounded by a polynomial in the block length. When applied to the Garcia-Stichtenoth tower, this yields codes of rate @math over an alphabet of size @math , that can correct a fraction of @math errors. Motivated by a problem of transmitting supplemental data over broadcast channels (Birk and Kol, INFOCOM 1998), we study the following coding problem: a sender communicates with n receivers R1,..., Rn. He holds an input x ∈ 0,01l n and wishes to broadcast a single message so that each receiver Ri can recover the bit xi. Each Ri has prior side information about x, induced by a directed graph Grain nodes; Ri knows the bits of a; in the positions j | (i,j) is an edge of G .G is known to the sender and to the receivers. We call encoding schemes that achieve this goal INDEXcodes for 0,1 n with side information graph G. In this paper we identify a measure on graphs, the minrank, which exactly characterizes the minimum length of linear and certain types of nonlinear INDEX codes. We show that for natural classes of side information graphs, including directed acyclic graphs, perfect graphs, odd holes, and odd anti-holes, minrank is the optimal length of arbitrary INDEX codes. For arbitrary INDEX codes and arbitrary graphs, we obtain a lower bound in terms of the size of the maximum acyclic induced subgraph. This bound holds even for randomized codes, but has been shown not to be tight. We study the approximate message-passing decoder for sparse superposition coding on the additive white Gaussian noise channel and extend our preliminary work. We use heuristic statistical-physics-based tools, such as the cavity and the replica methods, for the statistical analysis of the scheme. While superposition codes asymptotically reach the Shannon capacity, we show that our iterative decoder is limited by a phase transition similar to the one that happens in low density parity check codes. We consider two solutions to this problem, that both allow to reach the Shannon capacity: 1) a power allocation strategy and 2) the use of spatial coupling, a novelty for these codes that appears to be promising. We present, in particular, simulations, suggesting that spatial coupling is more robust and allows for better reconstruction at finite code lengths. Finally, we show empirically that the use of a fast Hadamard-based operator allows for an efficient reconstruction, both in terms of computational time and memory, and the ability to deal with very large messages. Independent sets play a central role in distributed algorithmics. We examine here the minimal requirements for computing non-trivial independent sets. In particular, we focus on algorithms that operate in a single communication round. A classic result of Linial shows that a constant number of rounds does not suffice to compute a maximal independent set. We are therefore interested in the size of the solution that can be computed, especially in comparison to the optimal. Our main result is a randomized one-round algorithm that achieves poly-logarithmic approximation on graphs of polynomially bounded-independence. Specifically, we show that the algorithm achieves the Caro-Wei bound (an extension of the Turan bound for independent sets) in general graphs up to a constant factor, and that the Caro-Wei bound yields a poly-logarithmic approximation on bounded-independence graphs. The algorithm uses only a single bit message and operates in a beeping model, where a node receives only the disjunction of the bits transmitted by its neighbors. We give limitation results that show that these are the minimal requirements for obtaining non-trivial solutions. In particular, a sublinear approximation cannot be obtained in a single round on general graphs, nor when nodes cannot both transmit and receive messages. We also show that our analysis of the Caro-Wei bound on polynomially bounded-independence graphs is tight, and that the poly-logarithmic approximation factor does not extend to @math O(1)-claw free graphs.