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Contemporary Coding Theory
Coding Theory naturally lies at the intersection of a large number
of disciplines in pure and applied mathematics. A multitude of
methods and means has been designed to construct, analyze, and
decode the resulting codes for communication. This has suggested to
bring together researchers in a variety of disciplines within
Mathematics, Computer Science, and Electrical Engineering, in order
to cross-fertilize generation of new ideas and force global
advancement of the field. Areas to be covered are Network Coding,
Subspace Designs, General Algebraic Coding Theory, Distributed
Storage and Private Information Retrieval (PIR), as well as
Code-Based Cryptography
AG codes achieve list decoding capacity over contant-sized fields
The recently-emerging field of higher order MDS codes has sought to unify a
number of concepts in coding theory. Such areas captured by higher order MDS
codes include maximally recoverable (MR) tensor codes, codes with optimal
list-decoding guarantees, and codes with constrained generator matrices (as in
the GM-MDS theorem).
By proving these equivalences, Brakensiek-Gopi-Makam showed the existence of
optimally list-decodable Reed-Solomon codes over exponential sized fields.
Building on this, recent breakthroughs by Guo-Zhang and Alrabiah-Guruswami-Li
have shown that randomly punctured Reed-Solomon codes achieve list-decoding
capacity (which is a relaxation of optimal list-decodability) over linear size
fields. We extend these works by developing a formal theory of relaxed higher
order MDS codes. In particular, we show that there are two inequivalent
relaxations which we call lower and upper relaxations. The lower relaxation is
equivalent to relaxed optimal list-decodable codes and the upper relaxation is
equivalent to relaxed MR tensor codes with a single parity check per column.
We then generalize the techniques of GZ and AGL to show that both these
relaxations can be constructed over constant size fields by randomly puncturing
suitable algebraic-geometric codes. For this, we crucially use the generalized
GM-MDS theorem for polynomial codes recently proved by Brakensiek-Dhar-Gopi. We
obtain the following corollaries from our main result. First, randomly
punctured AG codes of rate achieve list-decoding capacity with list size
and field size . Prior to this work, AG
codes were not even known to achieve list-decoding capacity. Second, by
randomly puncturing AG codes, we can construct relaxed MR tensor codes with a
single parity check per column over constant-sized fields, whereas
(non-relaxed) MR tensor codes require exponential field size.Comment: 38 page
Integer linear programming techniques for constant dimension codes and related structures
The lattice of subspaces of a finite dimensional vector space over a finite field is combined with the so-called subspace distance or the injection distance a metric space. A subset of this metric space is called subspace code. If a subspace code contains solely elements, so-called codewords, with equal dimension, it is called constant dimension code, which is abbreviated as CDC. The minimum distance is the smallest pairwise distance of elements of a subspace code. In the case of a CDC, the minimum distance is equivalent to an upper bound on the dimension of the pairwise intersection of any two codewords. Subspace codes play a vital role in the context of random linear network coding, in which data is transmitted from a sender to multiple receivers such that participants of the communication forward random linear combinations of the data. The two main problems of subspace coding are the determination of the cardinality of largest subspace codes and the classification of subspace codes. Using integer linear programming techniques and symmetry, this thesis answers partially the questions above while focusing on CDCs. With the coset construction and the improved linkage construction, we state two general constructions, which improve on the best known lower bound of the cardinality in many cases. A well-structured CDC which is often used as building block for elaborate CDCs is the lifted maximum rank distance code, abbreviated as LMRD. We generalize known upper bounds for CDCs which contain an LMRD, the so-called LMRD bounds. This also provides a new method to extend an LMRD with additional codewords. This technique yields in sporadic cases best lower bounds on the cardinalities of largest CDCs. The improved linkage construction is used to construct an infinite series of CDCs whose cardinalities exceed the LMRD bound. Another construction which contains an LMRD together with an asymptotic analysis in this thesis restricts the ratio between best known lower bound and best known upper bound to at least 61.6% for all parameters. Furthermore, we compare known upper bounds and show new relations between them. This thesis describes also a computer-aided classification of largest binary CDCs in dimension eight, codeword dimension four, and minimum distance six. This is, for non-trivial parameters which in addition do not parametrize the special case of partial spreads, the third set of parameters of which the maximum cardinality is determined and the second set of parameters with a classification of all maximum codes. Provable, some symmetry groups cannot be automorphism groups of large CDCs. Additionally, we provide an algorithm which examines the set of all subgroups of a finite group for a given, with restrictions selectable, property. In the context of CDCs, this algorithm provides on the one hand a list of subgroups, which are eligible for automorphism groups of large codes and on the other hand codes having many symmetries which are found by this method can be enlarged in a postprocessing step. This yields a new largest code in the smallest open case, namely the situation of the binary analogue of the Fano plane
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum