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The study of new error correcting codes has raised attention in the last years, especially because of their use in cryptosystems that are resistant to attacks running on quantum computers. In 2006, while leaving a more in-depth analysis for future research, Stakhov gave some interesting ideas on how to exploit Fibonacci numbers to derive an original error correcting code with a compact representation. In this paper, we provide an explicit formula to compute the redundancy of Stakhov codes, we identify some flows in the initial decoding procedure described by Stakhov, whose crucial point is to solve some non-trivial Diophantine equations, and provide a detailed discussion on how to avoid solving such equations in some cases and on how to detect and correct errors more efficiently. [ABSTRACT FROM AUTHOR]

We study locally recoverable codes on algebraic curves. In the first part of the manuscript, we provide a bound on the generalized Hamming weight of these codes. In the second part, we propose a new family of algebraic geometric LRC codes, which are LRC codes from the Norm-Trace curve. Finally, using some properties of Hermitian codes, we improve the bounds on the distance proposed in Barg et al. (2015) [1] of some Hermitian LRC codes. [ABSTRACT FROM AUTHOR]

In addition to their usefulness in proving one's identity electronically, identification protocols based on zero-knowledge proofs allow designing secure cryptographic signature schemes by means of the Fiat–Shamir transform or other similar constructs. This approach has been followed by many cryptographers during the NIST (National Institute of Standards and Technology) standardization process for quantum-resistant signature schemes. NIST candidates include solutions in different settings, such as lattices and multivariate and multiparty computation. While error-correcting codes may also be used, they do not provide very practical parameters, with a few exceptions. In this manuscript, we explored the possibility of using the error-correcting codes proposed by Stakhov in 2006 to design an identification protocol based on zero-knowledge proofs. We showed that this type of code offers a valid alternative in the error-correcting code setting to build such protocols and, consequently, quantum-resistant signature schemes. [ABSTRACT FROM AUTHOR]

In this paper we consider the Hermitian codes defined as the dual codes of one-point evaluation codes on the Hermitian curve H over the finite field F q 2 . We focus on those with distance d ≥ q 2 − q and give a geometric description of the support of their minimum-weight codewords. We consider the unique writing μ q + λ (q + 1) of the distance d with μ , λ non negative integers, and μ ≤ q , and consider all the curves X of the affine plane A F q 2 2 of degree μ + λ defined by polynomials with x μ y λ as leading monomial with respect to the DegRevLex term ordering (with y > x). We prove that a zero-dimensional subscheme Z of A F q 2 2 is the support of a minimum-weight codeword of the Hermitian code with distance d if and only if it is made of d simple F q 2 -points and there is a curve X such that Z coincides with the scheme theoretic intersection H ∩ X (namely, as a cycle, Z = H ⋅ X). Finally, exploiting this geometric characterization, we propose an algorithm to compute the number of minimum weight codewords and we present comparison tables between our algorithm and MAGMA command MinimumWords. [ABSTRACT FROM AUTHOR]