Algebraic List-decoding of Subspace Codes

Subspace codes were introduced in order to correct errors and erasures for randomized network coding, in the case where network topology is unknown (the noncoherent case). Subspace codes are indeed collections of subspaces of a certain vector space over a finite field. The Koetter-Kschischang construction of subspace codes are similar to Reed-Solomon codes in that codewords are obtained by evaluating certain (linearized) polynomials. In this paper, we consider the problem of list-decoding the Koetter-Kschischang subspace codes. In a sense, we are able to achieve for these codes what Sudan was able to achieve for Reed-Solomon codes. In order to do so, we have to modify and generalize the original Koetter-Kschischang construction in many important respects. The end result is this: for any integer $L$, our list-$L$ decoder guarantees successful recovery of the message subspace provided that the normalized dimension of the error is at most $L - \frac{L(L+1)}{2}R$ where $R$ is the normalized packet rate. Just as in the case of Sudan's list-decoding algorithm, this exceeds the previously best known error-correction radius $1-R$, demonstrated by Koetter and Kschischang, for low rates $R$

A New Algebraic Approach for String Reconstruction from Substring Compositions

We consider the problem of binary string reconstruction from the multiset of its substring compositions, i.e., referred to as the substring composition multiset, first introduced and studied by Acharya et al. We introduce a new algorithm for the problem of string reconstruction from its substring composition multiset which relies on the algebraic properties of the equivalent bivariate polynomial formulation of the problem. We then characterize specific algebraic conditions for the binary string to be reconstructed that guarantee the algorithm does not require any backtracking through the reconstruction, and, consequently, the time complexity is bounded polynomially. More specifically, in the case of no backtracking, our algorithm has a time complexity of $O(n^2)$ compared to the algorithm by Acharya et al., which has a time complexity of $O(n^2\log(n))$, where $n$ is the length of the binary string. Furthermore, it is shown that larger sets of binary strings are uniquely reconstructable by the new algorithm and without the need for backtracking leading to codebooks of reconstruction codes that are larger, by a linear factor in size, compared to the previously known construction by Pattabiraman et al., while having $O(n^2)$ reconstruction complexity