Optimal rate algebraic list decoding using narrow ray class fields

We use class field theory, specifically Drinfeld modules of rank 1, to construct a family of asymptotically good algebraic-geometric (AG) codes over fixed alphabets. Over a field of size $\ell^2$, these codes are within $2/(\sqrt{\ell}-1)$ of the Singleton bound. The functions fields underlying these codes are subfields with a cyclic Galois group of the narrow ray class field of certain function fields. The resulting codes are "folded" using a generator of the Galois group. This generalizes earlier work by the first author on folded AG codes based on cyclotomic function fields. Using the Chebotarev density theorem, we argue the abundance of inert places of large degree in our cyclic extension, and use this to devise a linear-algebraic algorithm to list decode these folded codes up to an error fraction approaching $1-R$ where $R$ is the rate. The list decoding can be performed in polynomial time given polynomial amount of pre-processed information about the function field. Our construction yields algebraic codes over constant-sized alphabets that can be list decoded up to the Singleton bound --- specifically, for any desired rate $R \in (0,1)$ and constant \eps > 0, we get codes over an alphabet size (1/\eps)^{O(1/\eps^2)} that can be list decoded up to error fraction 1-R-\eps confining close-by messages to a subspace with N^{O(1/\eps^2)} elements. Previous results for list decoding up to error-fraction 1-R-\eps over constant-sized alphabets were either based on concatenation or involved taking a carefully sampled subcode of algebraic-geometric codes. In contrast, our result shows that these folded algebraic-geometric codes {\em themselves} have the claimed list decoding property.Comment: 22 page

Efficiently list-decodable punctured Reed-Muller codes

The Reed-Muller (RM) code encoding $n$-variate degree-$d$ polynomials over ${\mathbb F}_q$ for $d < q$, with its evaluation on ${\mathbb F}_q^n$, has relative distance $1-d/q$ and can be list decoded from a $1-O(\sqrt{d/q})$ fraction of errors. In this work, for $d \ll q$, we give a length-efficient puncturing of such codes which (almost) retains the distance and list decodability properties of the Reed-Muller code, but has much better rate. Specificially, when $q =\Omega( d^2/\epsilon^2)$, we given an explicit rate $\Omega\left(\frac{\epsilon}{d!}\right)$ puncturing of Reed-Muller codes which have relative distance at least $(1-\epsilon)$ and efficient list decoding up to $(1-\sqrt{\epsilon})$ error fraction. This almost matches the performance of random puncturings which work with the weaker field size requirement $q= \Omega( d/\epsilon^2)$. We can also improve the field size requirement to the optimal (up to constant factors) $q =\Omega( d/\epsilon)$, at the expense of a worse list decoding radius of $1-\epsilon^{1/3}$ and rate $\Omega\left(\frac{\epsilon^2}{d!}\right)$. The first of the above trade-offs is obtained by substituting for the variables functions with carefully chosen pole orders from an algebraic function field; this leads to a puncturing for which the RM code is a subcode of a certain algebraic-geometric code (which is known to be efficiently list decodable). The second trade-off is obtained by concatenating this construction with a Reed-Solomon based multiplication friendly pair, and using the list recovery property of algebraic-geometric codes.Comment: 14 pages, To appear in IEEE Transactions on Information Theor

The asymptotic behavior of automorphism groups of function fields over finite fields

The purpose of this paper is to investigate the asymptotic behavior of automorphism groups of function fields when genus tends to infinity. Motivated by applications in coding and cryptography, we consider the maximum size of abelian subgroups of the automorphism group \mbox{Aut}(F/\mathbb{F}_q) in terms of genus ${g_F}$ for a function field $F$ over a finite field $\mathbb{F}_q$. Although the whole group \mbox{Aut}(F/\mathbb{F}_q) could have size $\Omega({g_F}^4)$, the maximum size $m_F$ of abelian subgroups of the automorphism group \mbox{Aut}(F/\mathbb{F}_q) is upper bounded by $4g_F+4$ for $g_F\ge 2$. In the present paper, we study the asymptotic behavior of $m_F$ by defining $M_q=\limsup_{{g_F}\rightarrow\infty}\frac{m_F \cdot \log_q m_F}{{g_F}}$, where $F$ runs through all function fields over $\mathbb{F}_q$. We show that $M_q$ lies between $2$ and $3$ (or $4$) for odd characteristic (or for even characteristic, respectively). This means that $m_F$ grows much more slowly than genus does asymptotically. The second part of this paper is to study the maximum size $b_F$ of subgroups of \mbox{Aut}(F/\mathbb{F}_q) whose order is coprime to $q$. The Hurwitz bound gives an upper bound $b_F\le 84(g_F-1)$ for every function field $F/\mathbb{F}_q$ of genus $g_F\ge 2$. We investigate the asymptotic behavior of $b_F$ by defining ${B_q}=\limsup_{{g_F}\rightarrow\infty}\frac{b_F}{{g_F}}$, where $F$ runs through all function fields over $\mathbb{F}_q$. Although the Hurwitz bound shows ${B_q}\le 84$, there are no lower bounds on $B_q$ in literature. One does not even know if ${B_q}=0$. For the first time, we show that ${B_q}\ge 2/3$ by explicitly constructing some towers of function fields in this paper

Efficient List-Decoding with Constant Alphabet and List Sizes

We present an explicit and efficient algebraic construction of capacity-achieving list decodable codes with both constant alphabet and constant list sizes. More specifically, for any $R \in (0,1)$ and $\epsilon>0$, we give an algebraic construction of an infinite family of error-correcting codes of rate $R$, over an alphabet of size $(1/\epsilon)^{O(1/\epsilon^2)}$, that can be list decoded from a $(1-R-\epsilon)$-fraction of errors with list size at most $\exp(\mathrm{poly}(1/\epsilon))$. Moreover, the codes can be encoded in time $\mathrm{poly}(1/\epsilon, n)$, the output list is contained in a linear subspace of dimension at most $\mathrm{poly}(1/\epsilon)$, and a basis for this subspace can be found in time $\mathrm{poly}(1/\epsilon, n)$. Thus, both encoding and list decoding can be performed in fully polynomial-time $\mathrm{poly}(1/\epsilon, n)$, except for pruning the subspace and outputting the final list which takes time $\exp(\mathrm{poly}(1/\epsilon))\cdot\mathrm{poly}(n)$. Our codes are quite natural and structured. Specifically, we use algebraic-geometric (AG) codes with evaluation points restricted to a subfield, and with the message space restricted to a (carefully chosen) linear subspace. Our main observation is that the output list of AG codes with subfield evaluation points is contained in an affine shift of the image of a block-triangular-Toeplitz (BTT) matrix, and that the list size can potentially be reduced to a constant by restricting the message space to a BTT evasive subspace, which is a large subspace that intersects the image of any BTT matrix in a constant number of points. We further show how to explicitly construct such BTT evasive subspaces, based on the explicit subspace designs of Guruswami and Kopparty (Combinatorica, 2016), and composition