Deletion correcting codes meet the Littlewood-Offord problem (Q2205891)

Let \(n, k\in\mathbb{Z}^+, b \in\mathbb{Z}_n\). The Levenshtein code \(L_b(k, n)\) is the set of all binary \(k\)-tuples \(\langle x_1, \ldots, x_k\rangle\) such that \(\sum\limits_{i=1\mathrm{div} k}ix_i\equiv b\pmod n\). The Helberg code \(H_b(k, s)\) is the set of all binary \(k\)-tuples \(\langle x_1, \ldots, x_k\rangle\) such that \(\sum_{i=1}^k v_ix_i\equiv b\pmod n\), where \(v_i=0\) for \(i\leq 0\), \(v_i=1+\sum\limits_{j=1}^s v_{i-j},\) for \(i\geq 1\), \(n=v_{k+1}\) and \(b\in\mathbb{Z}_n.\) The celebrated Littlewood-Offord problem concerning the concentration of subset sums for complex numbers asks to determine or estimate the number of sums of the form \(\varepsilon_1z_1 +\cdots + \varepsilon_kz_k\) that fall in the interior of a given disc of unit radius, where \(z_i\)'s are complex numbers of modulus at least one and each \(i\) takes the value \(1\) or \(-1\). In this paper, the authors show that a result on the Littlewood-Offord problem gives an upper bound for the size of the Levenshtein code and the Helberg code. Also, in the last section of the paper it is shown that a recent result on the deletion correcting codes gives a modular analogue of the Littlewood-Offord problem which generalizes some previously known results.