A Family of Codes With Locality Containing Optimal Codes
Victor Gonzalo Lopez NeumannIEEE Access
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Abstract
Locally recoverable codes were introduced by Gopalan et al. in 2012, and in the same year Prakash et al. introduced the concept of codes with locality, which are a type of locally recoverable codes. In this work we introduce a new family of codes with locality, which are subcodes of a certain family of evaluation codes. We determine the dimension of these codes, and also bounds for the minimum distance. We present the true values of the minimum distance in special cases, and also show that some elements of this family are ''optimal codes'', as defined by Prakash et al. INDEX TERMS Locally recoverable codes, affine cartesian codes. II. A FAMILY OF LOCALLY RECOVERABLE CODES Let F q be a finite field with q elements. Definition 2.1: Let m, r, δ be positive integers, with δ ≥ 2 and r +δ−1 ≤ m. We say that a (linear) code C ⊂ F m q is (r, δ)locally recoverable if for every i ∈ {1,. .. , m} there exists a subset S i ⊂ {1,. .. , m}, containing i and of cardinality at most r + δ − 1, such that the punctured code obtained by removing the entries which are not in S i has minimum distance at least δ. The condition on the minimum distance in the above definition shows that one cannot have two distinct codewords in the punctured code which coincide in (at least) r positions,
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My postgraduate experience has been fruitful and exciting, thanks to the people that have accompanied me through this journey. They have played a significant role in molding me into who I am today. Firstly, I would like to express my heartfelt gratitude to my supervisor Prof. Chee Yeow Meng for his guidance and support throughout my graduate studies. I was fortunate to have the opportunity to work with him in the field of coding theory. His vast knowledge of this field gave me great insights in my own research. I am also grateful for his sound advice and encouragement whenever I was facing difficulties. I am greatly appreciative of my friend and mentor, Kiah Han Mao. He had on many occasions shared his academic experience with me and always offered precious advice in my academic path. It was a pleasure discussing and collaborating with him and I am grateful for his help on my research. I am deeply grateful to my friend and colleague, Punarbasu. He has always been generous in sharing his knowledge and offering a helping hand in obstacles that I faced. I learnt many invaluable lessons from him both academically and in life. I would also like to thank my friends, peers and professors from NTU. I have enjoyed the companionship of friends which made learning more interesting, peers who contributed to a positive work environment and professors who are always friendly and approachable in sharing their wisdom. Last but not least, I am truly blessed to have the love, support and encouragement from my beloved family and my adorable girlfriend, Khai Ting. They have been my pillar of strength in this journey and continuously motivates me to strive for the better.
A new family of locally correctable codes based on degree-lifted algebraic geometry codes
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We describe new constructions of error correcting codes, obtained by "degree-lifting" a short algebraic geometry (AG) base-code of block-length q to a lifted-code of block-length q m , for arbitrary integer m. The construction generalizes the way degree-d, univariate polynomials evaluated over the q-element field (also known as Reed-Solomon codes) are "lifted" to degree-d, m-variate polynomials (Reed-Muller codes). Three properties are established: Rate The rate of the degree-lifted code is approximately a 1 m!-fraction of the rate of the basecode. Distance The relative distance of the degree-lifted code is at least as large as that of the basecode. This is proved using a generalization of the Schwartz-Zippel Lemma to degree-lifted Algebraic-Geometry codes (Lemma 5.6). As a first concrete example, lifting the AG codes of Garcia and Stichtenoth [J. Number Theory '96] results in Reed-Muller-like codes but over constant sized alphabets, such codes are interesting in the search for probabilistically checkable proofs (PCPs) with constant rate and polynomially small query complexity. Local correction If the base code is invariant under a group that is "close" to being doublytransitive (in a precise manner defined later , cf. Definition 6.1) then the degree-lifted code is locally correctable with query complexity at most q 2. The automorphisms of the base-code are crucially used to generate query-sets, abstracting the use of affine-lines in the local correction procedure of Reed-Muller codes. Taking a second concrete example, we show that degree-lifted Hermitian codes form a family of locally correctable codes over an alphabet that is significantly smaller than that obtained by Reed-Muller codes of similar constant rate, message length, and distance.
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