We present a modified version of the Bravyi-Terhal bound that applies to quantum codes defined by local parity-check constraints on aD-dimensional lattice quotient. Specifically, we consider a quotient ZD/Λ of ZDof cardinality ℓ, where Λ is someD-dimensional sublattice of ZD: we suppose that every vertex of this quotient indexesmqubits of a stabilizer codeC, which therefore has lengthn=mℓ. We prove that if all stabilizer generators act on qubits whose indices lie within a ball of radius ρ, then the minimum distancedof the code satisfiesd≤m√ γD( √D+ 4ρ)ℓD−1/D, where γDis theD-dimensional Hermite constant. We then apply this bound to derive an upper bound on the minimum distance of Abelian Two-Block Group Algebra (2BGA) codes whose parity-check matrices have the form [A|B] with each submatrix representing an element of a group algebra over a finite abelian group.

Generalized Bicycle (GB) codes offer a compelling alternative to surface codes for quantum error correction. This paper focuses on (2,2)-Generalized Bicycle codes, constructed from pairs of binary circulant matrices with two non-zero elements per row. Leveraging a lower bound on their minimum distance, we construct three novel infinite families of optimal (2,2)-GB codes with parameters [[2n2, 2,n]], [[4r2, 2, 2r]], and [[(2t+1)2+1, 2, 2t+1]]. These families match the performance of Kitaev’s toric code and the best 2D weight-4 surface codes, reaching known theoretical limits. In particular, the second family breaks a long-held belief by providing optimal even-distance GB codes, previously deemed impossible. All are CSS codes derived from Cayley graphs. Recognizing that standard equivalence relations do not preserve their CSS structure, we introduce a CSS-preserving equivalence relation for rigorous comparison of Cayley graph-based CSS codes. Under this framework, the first two families are inequivalent to all previously known optimal weight-4 2D surface codes, while the third family is equivalent to the best-known odd-distance 2D surface code. Finally, we classify all extremal, non-equivalent (2, 2)-GB codes with length below 200 and present a comparison table with existing notable 2D weight-4 surface codes.

Linear finite state machines (LFSMs) are particular primitives widely used in information theory, coding theory and cryptography. Among those linear automata, a particular case of study is linear feedback shift registers (LFSRs) used in many cryptographic applications such as design of stream ciphers or pseudo-random generation. LFSRs could be seen as particular LFSMs without inputs. In this paper, we first recall the description of LFSMs using traditional matrices representation. Then, we introduce a new matrices representation with polynomial fractional coefficients. This new representation leads to sparse representations and implementations. As direct applications, we focus our work on the Windmill generators case, used for example in the E0 stream cipher and on other general applications that use this new representation. In a second part, a new design criterion called diffusion delay for LFSRs is introduced and well compared with existing related notions. This criterion represents the diffusion capacity of an LFSR. Thus, using the matrices representation, we present a new algorithm to randomly pick LFSRs with good properties (including the new one) and sparse descriptions dedicated to hardware and software designs. We present some examples of LFSRs generated using our algorithm to show the relevance of our approach.

This paper corrects some errors on a previous paper concerning the synthesis of Feedback with Carry Shift Registers using the Euclidean Algorithm.

This article describes new theoretical results concerning the general behavior of a feedback with carry shift register (FCSR) automaton. They help to better understand how the initial parameters must be chosen to use this automaton as a basic block of a filtered stream cipher. These results especially concern the structure of the transition graph of an FCSR automaton and the number of iterations of the FCSR transition function required to reach the main part of the graph. A potential linear weakness and a easy way to prevent the corresponding attack are also given.

Feedback with carry shift registers (FCSR) was introduced by Goresky and Klapper in 1993. It is similar to the classical linear feedback shift registers (LFSR) used in many pseudorandom generators. The main difference is that the elementary additions are not additions modulo 2 but with propagation of carries. The main problem for the use of an FCSR automaton is the fact that the generated sequences are predictable. In order to remove this weakness of FCSR-based generators, we propose filtering the state of the FCSR with a linear function. This method is efficient since the FCSR structure is not related to a linear property. This paper presents an extensive study of FCSR automata, a security analysis of our generator (concerning linear and 2-adic cryptanalysis, algebraic attack, correlation attack, etc.), and a practical example of parameters in order to design this generator. An important point concerning this generator is the fact that it is simple and efficient, both in hardware and software implementation.

Feedback with carry shift registers (FCSR) were introduced by Klapper and Goresky (1994). They are very similar to classical linear feedback shift registers (LFSR) used in many pseudorandom generators. The main difference is the fact that the elementary additions are not additions modulo 2 but with propagation of carries. The mathematical models for LFSR are equivalently linear recurring sequences over GF(2) or rational series in the set GF(2)[[x]]. For FCSR, the "good" model is the one of rational 2-adic numbers. It is well known, that the series generated by a LFSR can be synthesized by either the Berlekamp-Massey algorithm for binary linear recurring sequences or the extended Euclidean algorithm in the set GF(2)[x] of binary polynomials. Klapper and Goresky (1997) give an algorithm for the FCSR synthesis. This algorithm is similar to those of Berlekamp-Massey and is based on De Weger and Mahler's rational approximation theory. In this correspondence, we prove that it is possible to synthesize the FCSR with the extended Euclidean algorithm in the ring /spl Zopf/ of integers. This algorithm is clearly equivalent to the previous one, however, it is simpler to understand, to implement, and to prove. Our algorithm is still valid in the case of g-adic integers where g is a positive integer. We also give a near-adaptative version of this algorithm.