Alberto Ravagnani

Johan Vester Dinesen, Ragnar Freij-Hollanti, Camilla Hollanti et al.

We consider the problem of determining the expected dimension of the star product of two uniformly random linear codes that are not necessarily of the same dimension. We use a correspondence between the star product and the evaluation of bilinear forms to provide an explicit lower bound on the expected star product dimension. We prove that the expected dimension asymptotically reaches its maximum possible value as the field size increases. Furthermore, we show that the same maximal dimension is achieved asymptotically as the code dimensions increase, subject to a condition bounding their relative growth rates. We also analyze the variance of the star product dimension, providing explicit asymptotic upper bounds. Finally, we discuss the implications of these results for private information retrieval, secure distributed matrix multiplication, quantum error correction, and cryptanalysis.

Altan B. Kilic, Alberto Ravagnani, Flavio Salizzoni

We consider the problem of computing the minimum length of functional batch and PIR codes of fixed dimension and for a fixed list size, over an arbitrary finite field. We recover, generalize, and refine several results that were previously obtained for binary codes. We present new upper and lower bounds for the minimum length, and discuss the asymptotic behaviour of this parameter. We also compute its value for several parameter sets. The paper also offers insights into the "correct" list size to consider for the Functional Batch Conjecture over non-binary finite fields, and establishes various supporting results.

Matteo Bertuzzo, Alberto Ravagnani, Eitan Yaakobi

The coverage depth problem in DNA data storage is about computing the expected number of reads needed to recover all encoded strands. Given a generator matrix of a linear code, this quantity equals the expected number of randomly drawn columns required to obtain full rank. While MDS codes are optimal when they exist, i.e., over large fields, practical scenarios may rely on structured code families defined over small fields. In this work, we develop combinatorial tools to solve the DNA coverage depth problem for various linear codes, based on duality arguments and the notion of extended weight enumerator. Using these methods, we derive closed formulas for the simplex, Hamming, ternary Golay, extended ternary Golay, and first-order Reed-Muller codes. The centerpiece of this paper is a general expression for the coverage depth of a linear code in terms of the weight distributions of its higher-field extensions.