Designs, Codes and Cryptography (2021) 89:1787–1837 https://doi.org/10.1007/s10623-021-00889-1 Hamming weight distributions of multi-twisted codes over finite fields Varsha Chauhan 1 · Anuradha Sharma 1 · Sandeep Sharma 1 · Monika Yadav 1 Received: 30 June 2020 / Revised: 28 January 2021 / Accepted: 4 May 2021 / Published online: 24 May 2021 © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021 Abstract Let F q denote the finite field of order q , and let n = m 1 +m 2 +· · ·+m ℓ , where m 1 , m 2 ,..., m ℓ are arbitrary positive integers (not necessarily coprime to q ). In this paper, we explicitly determine Hamming weights of all non-zero codewords of several classes of multi-twisted codes of length n and block lengths (m 1 , m 2 ,..., m ℓ ) over F q . As an application of these results, we explicitly determine Hamming weight distributions of several classes of multi- twisted codes of length n and block lengths (m 1 , m 2 ,..., m ℓ ) over F q . Among these classes of multi-twisted codes, we identify two classes of optimal equidistant linear codes that have nice connections with the theory of combinatorial designs and several other classes of minimal linear codes that are useful in constructing secret sharing schemes with nice access structures. We illustrate our results with some examples, and list many optimal, projective and minimal linear codes belonging to these classes of multi-twisted codes. Keywords Gauss sums · Few weight codes · Equidistant codes Mathematics Subject Classification 94B15 1 Introduction A linear code C of length n over F q is defined as an F q -linear subspace of F n q , where F q is the finite field with q elements, n is a positive integer and F n q is the n-dimensional vector space consisting of all n-tuples over F q . Elements of the code C are called codewords. The dimension k of the code C is defined as the dimension of C as an F q -linear subspace of F n q . A generator matrix of the code C is defined as a k × n matrix G, whose rows form a basis of the Communicated by C. Ding. V. Chauhan: Research support by UGC, India, is gratefully acknowledged. A. Sharma: Research support by DST-SERB, India, under Grant No. MTR/2017/000358 is gratefully acknowledged. S. Sharma: Research support by UGC, India, is gratefully acknowledged, M. Yadav: Research support by CSIR, India, is gratefully acknowledged. B Anuradha Sharma anuradha@iiitd.ac.in 1 Department of Mathematics, IIIT-Delhi, New Delhi 110020, India 123