On the equivalence of linear cyclic and constacyclic codes

We introduce new sufficient conditions for permutation and monomial equivalence of linear cyclic codes over various finite fields. We recall that monomial equivalence and isometric equivalence are the same relation for linear codes over finite fields. A necessary and sufficient condition for the monomial equivalence of linear cyclic codes through a shift map on their defining set is also given. Moreover, we provide new algebraic criteria for the monomial equivalence of constacyclic codes over $\mathbb{F}_4$. Finally, we prove that if $\gcd(3n,\phi(3n))=1$, then all permutation equivalent constacyclic codes of length $n$ over $\mathbb{F}_4$ are given by the action of multipliers. The results of this work allow us to prune the search algorithm for new linear codes and discover record-breaking linear and quantum codes.Comment: 18 page

Symplectic Groups, Symplectic Spreads, Codes, and Unimodular Lattices

AbstractIt is known that the symplectic groupSp2n(p) has two (complex conjugate) irreducible representations of degree (pn+1)/2 realized overQ(−p), provided thatp≡3mod4. In the paper we give an explicit construction of an odd unimodularSp2n(p)·2-invariant lattice Δ(p,n) in dimensionpn+1 for anypn≡3mod4. Such a lattice has been constructed by R. Bacher and B. B. Venkov in the casepn=27. A second main result says that these lattices are essentially unique. We show that forn≥3 the minimum of Δ(p,n) is at least (p+1)/2 and at mostp(n−1)/2. The interrelation between these lattices, symplectic spreads of Fp2n, and self-dual codes over Fpis also investigated. In particular, using new results of U. Dempwolff and L. Bader, W. M. Kantor, and G. Lunardon, we come to three extremal self-dual ternary codes of length 28

Dynamical Systems on Spectral Metric Spaces

Let (A,H,D) be a spectral triple, namely: A is a C*-algebra, H is a Hilbert space on which A acts and D is a selfadjoint operator with compact resolvent such that the set of elements of A having a bounded commutator with D is dense. A spectral metric space, the noncommutative analog of a complete metric space, is a spectral triple (A,H,D) with additional properties which guaranty that the Connes metric induces the weak*-topology on the state space of A. A *-automorphism respecting the metric defined a dynamical system. This article gives various answers to the question: is there a canonical spectral triple based upon the crossed product algebra AxZ, characterizing the metric properties of the dynamical system ? If $\alpha$ is the noncommutative analog of an isometry the answer is yes. Otherwise, the metric bundle construction of Connes and Moscovici is used to replace (A,$\alpha$) by an equivalent dynamical system acting isometrically. The difficulties relating to the non compactness of this new system are discussed. Applications, in number theory, in coding theory are given at the end