Minimum-weight codewords of the Hermitian codes are supported on complete intersections

Let $\mathcal{H}$ be the Hermitian curve defined over a finite field $\mathbb{F}_{q^2}$. In this paper we complete the geometrical characterization of the supports of the minimum-weight codewords of the algebraic-geometry codes over $\mathcal{H}$, started in [1]: if $d$ is the distance of the code, the supports are all the sets of $d$ distinct $\mathbb{F}_{q^2}$-points on $\mathcal{H}$ complete intersection of two curves defined by polynomials with prescribed initial monomials w.r.t. \texttt{DegRevLex}. For most Hermitian codes, and especially for all those with distance $d\geq q^2-q$ studied in [1], one of the two curves is always the Hermitian curve $\mathcal{H}$ itself, while if $d<q$ the supports are complete intersection of two curves none of which can be $\mathcal{H}$. Finally, for some special codes among those with intermediate distance between $q$ and $q^2-q$, both possibilities occur. We provide simple and explicit numerical criteria that allow to decide for each code what kind of supports its minimum-weight codewords have and to obtain a parametric description of the family (or the two families) of the supports. [1] C. Marcolla and M. Roggero, Hermitian codes and complete intersections, arXiv preprint arXiv:1510.03670 (2015)

The small weight codewords of the functional codes associated to non-singular hermitian varieties

This article studies the small weight codewords of the functional code C (Herm) (X), with X a non-singular Hermitian variety of PG(N, q (2)). The main result of this article is that the small weight codewords correspond to the intersections of X with the singular Hermitian varieties of PG(N, q (2)) consisting of q + 1 hyperplanes through a common (N - 2)-dimensional space I , forming a Baer subline in the quotient space of I . The number of codewords having these small weights is also calculated. In this way, similar results are obtained to the functional codes C (2)(Q), Q a non-singular quadric (Edoukou et al., J. Pure Appl. Algebra 214:1729-1739, 2010), and C (2)(X), X a non-singular Hermitian variety (Hallez and Storme, Finite Fields Appl. 16:27-35, 2010)

Cayley-Bacharach and evaluation codes on complete intersections

In recent work, J. Hansen uses cohomological methods to find a lower bound for the minimum distance of an evaluation code determined by a reduced complete intersection in the projective plane. In this paper, we generalize Hansen's results from P^2 to P^m; we also show that the hypotheses in Hansen's work may be weakened. The proof is succinct and follows by combining the Cayley-Bacharach theorem and bounds on evaluation codes obtained from reduced zero-schemes.Comment: 10 pages. v2: minor expository change

Intersections of the Hermitian surface with irreducible quadrics in PG(3,q2)PG(3,q^2), qq odd

In $PG(3,q^2)$, with $q$ odd, we determine the possible intersection sizes of a Hermitian surface $\mathcal{H}$ and an irreducible quadric $\mathcal{Q}$ having the same tangent plane $\pi$ at a common point $P\in{\mathcal Q}\cap{\mathcal H}$.Comment: 14 pages; clarified the case q=

Intersections of the Hermitian Surface with irreducible Quadrics in even Characteristic

We determine the possible intersection sizes of a Hermitian surface $\mathcal H$ with an irreducible quadric of ${\mathrm PG}(3,q^2)$ sharing at least a tangent plane at a common non-singular point when $q$ is even.Comment: 20 pages; extensively revised and corrected version. This paper extends the results of arXiv:1307.8386 to the case q eve

Intersection sets, three-character multisets and associated codes

In this article we construct new minimal intersection sets in ${\mathrm{AG}}(r,q^2)$ sporting three intersection numbers with hyperplanes; we then use these sets to obtain linear error correcting codes with few weights, whose weight enumerator we also determine. Furthermore, we provide a new family of three-character multisets in ${\mathrm{PG}}(r,q^2)$ with $r$ even and we also compute their weight distribution.Comment: 17 Pages; revised and corrected result

Quantum Error Correcting Codes From The Compression Formalism

We solve the fundamental quantum error correction problem for bi-unitary channels on two-qubit Hilbert space. By solving an algebraic compression problem, we construct qubit codes for such channels on arbitrary dimension Hilbert space, and identify correctable codes for Pauli-error models not obtained by the stabilizer formalism. This is accomplished through an application of a new tool for error correction in quantum computing called the ``higher-rank numerical range''. We describe its basic properties and discuss possible further applications.Comment: 8 pages, 2 figures, Rep. Math. Phys., to appea