Commutative algebra and coding theory, with applications to quantum error-correction

• Autores: Rodrigo San José Rubio
• Directores de la Tesis: Philippe Thiery Giménez Martín (dir. tes.) , Diego Ruano Benito (codir. tes.)
• Lectura: En la Universidad de Valladolid ( España ) en 2024
• Idioma: inglés
• Tribunal Calificador de la Tesis: Sudhir Ghorpade (presid.) , Eduardo Sáenz de Cabezón Irigaray (secret.) , Jade Nardi (voc.)
• Enlaces
  • Tesis en acceso abierto en: UVADOC
• Resumen

  • Algebraic Coding Theory plays an important role not only in many different aspects of communication but also in cryptography and quantum computing. In this thesis we are interested in using tools from Commutative Algebra to derive properties of linear codes. We focus mainly on evaluation codes, since they have a natural connection to Commutative Algebra, but we also consider other types of codes such as cyclic codes (which can be viewed as subfield subcodes of evaluation codes) or matrix-product codes.

    Many aspects of evaluation codes can be understood by means of the vanishing ideal of the set of points considered. A natural question that arises is how to compute this vanishing ideal. In the projective setting one usually has to compute the radical of an ideal. We give an alternative and more efficient way of computing the vanishing ideal by using the saturation with respect to the homogeneous maximal ideal. Another option to study evaluation codes over the projective space is to consider a set of fixed representatives of the points, regarded as a subset of the affine space, and its vanishing ideal. We give a universal Gröbner basis for this vanishing ideal when the set of points corresponds to certain subsets of the projective line, or to the whole projective space.

    Obtaining long codes with good parameters over a small finite field, which is desirable for applications, is a complicated problem in general. One approach to achieve this is to consider subfield subcodes. We obtain bases for the subfield subcodes of projective Reed-Solomon codes and projective Reed-Muller codes in many cases. We also give an alternative approach using a recursive construction for projective Reed-Muller codes.

    The interest of the generalized Hamming weights of a linear code originates from the fact that they determine its performance on the wire-tap channel of type II, but many other applications have been found for them since then. We provide lower and upper bounds for the generalized Hamming weights of projective Reed-Muller codes, determining the true values in many cases. We also provide bounds for the generalized Hamming weights of matrix-product codes. As a sample of our results, we obtain the exact value of the generalized Hamming weights of matrix-product codes obtained with two Reed-Solomon codes.

    The development of reliable quantum computing and communication requires error-correction to deal with noise and decoherence. To perform error-correction, we can consider stabilizer quantum codes. The CSS construction provides a way to construct such codes using a pair of classical linear codes. The dimension of the relative hull of this pair of classical codes gives the minimum number required of maximally entangled pairs. We have used the subfield subcodes of projective Reed-Solomon codes to construct both symmetric and asymmetric entanglement-assisted quantum error-correcting codes. Moreover, we have computed the hulls of projective Reed-Muller codes over the projective plane, determining all the parameters of the corresponding quantum codes. We also study how to change the dimension of the hull of projective Reed-Muller codes by considering monomially equivalent codes, giving rise to families of codes with a flexible amount of entanglement.

    One of the main problems for quantum computing is the fault-tolerant implementation of non-Clifford gates. We study CSS-T codes, which are quantum codes derived from the CSS construction that support the transversal T gate. We give a new characterization of CSS-T codes, and we use it to determine which CSS-T codes can be constructed from cyclic codes. Moreover, we also obtain a propagation rule for nondegenerate CSS-T codes, and we use it to obtain CSS-T codes with better parameters than those available in the literature.