Intertwined Results on Linear Codes and Galois Geometries
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Intertwined Results on Linear Codes and Galois Geometries Peter Vandendriessche promotor: prof. dr. Leo Storme ii Contents 1 Preliminaries: Finite Geometry & Coding Theory 7 1.1 Finite geometry . .7 1.2 Coding theory . 10 2 LDPC codes derived from Galois geometries 13 2.1 Motivation and preliminaries . 13 2.2 LDPC codes from projective and affine geometries . 15 2.3 LDPC codes from linear representations . 21 2.4 LDPC codes from Hermitian varieties . 32 2.5 LDPC codes from partial geometries . 48 3 (q + t; t)-arcs of type (0; 2; t) 55 3.1 Preliminaries and motivation . 55 3.2 A basis for PG(2; q), q even............................ 56 3.3 Projective triads and (q + t; t)-arcs of type (0; 2; t)............... 59 3.4 A new infinite family . 61 4 Optimal blocking multisets 67 4.1 Preliminaries and motivation . 67 4.2 A new way of looking: rational sums of hyperplanes . 70 4.3 Generalizations of previous results . 73 4.4 A surprising new link with coding theory . 76 iii iv CONTENTS 5 Small line sets with few odd-points 81 5.1 Motivation and preliminaries . 81 5.2 The affine case . 83 5.3 The projective case . 88 6 Geometries over finite chain rings 93 6.1 Motivation and preliminaries . 93 6.2 Standard form representation of modules . 97 6.3 Extension of Kantor's theorem to finite chain rings . 101 7 Miscellaneous results 109 7.1 Generalizing AM-GM and Turkevich's inequality . 109 7.2 Large weight code words for PG(n; q)...................... 114 7.3 Blocking sets of the Hermitian unital . 122 A Building a low-cost GPU based supercomputer 135 A.1 The hardware . 135 A.2 Efficient parallel computing . 138 A.3 The LDPC decoding algorithm . 147 A.4 An OpenCL implementation . 157 Summary The curious interweaving of Galois geometry and linear codes has been inspiring researchers for decades to explore the many intersections of these two topics. This interweaving is also the core theme of this PhD thesis, in which I will be presenting results which all in a sense revolve around the intertwining of linear codes and Galois geometries. Chapter 1 repeats the general notations in finite geometry and coding theory. This chapter introduces the notations used throughout this thesis. Chapter 2 is about (LDPC) codes derived from finite geometries. In Section 2.1, I discuss the motivation to study LDPC codes from finite geometries, as well as provide some general preliminaries on this topic, which appear in several of the sections afterwards. In Section 2.2, I discuss the cyclic LDPC codes constructed from affine and projective geome- tries, as this is the most well-known class of finite geometry LDPC codes. In this section I also present some results which, even though they look like they should have been discovered decades ago, I have not been able to find in the literature, so which I will assume to be new. In particular, we show that the order of this code as a cyclic code is equal to the length of the code, and we study what happens when removing the all-one vector from the code in a canonical way, resulting in an additional application of my earlier paper [94] with J. Limbu- pasiriporn and L. Storme, which will be discussed in Section 7.2. The results presented in this section are (a small) part of joint work with Y. Fujiwara, which is submitted to IEEE Trans. Inform. Theory [45]. In Section 2.3, I discuss the LDPC codes from points and lines in linear representations ∗ T2 (K). My first results on this topic date back to before my official research started, as a spin-off of an assignment during my bachelor's studies. Here, I showed that when K is an arc and the characteristic of the field of the code is different from that of the field of the underlying geometry, and another condition on the finite field is met, then the code is entirely generated by its code words of minimum weight. Moreover, from this I could compute the actual dimension of the code. Earlier work by Pepe et al. [113] demonstrated this behavior only up to a certain small Hamming weight, and needed in some cases two types of small weight code words to generate them, instead of one (hence my result implies that code words of their second type are also a linear combination of code words of their first type). This result was published in Des. Codes Cryptogr. [139]. Unfortunately, the \another condition on the finite field” excluded the important case of binary codes. In the special case that K is a conic minus one point, P. Sin and Q. Xiang showed the dimension formula for the case of binary 1 2 CONTENTS codes [124]. Later, I managed to improve my technique and show that when the characteristic of the field of the code is different from that of the field of the underlying geometry, then the code is entirely generated by its code words of minimum weight, and the same dimension formula remains valid in this more general setting. This new result improves on all of the previous results: it generalized my own result from [140], it embedded the formula from [124] in a large infinite class of geometries and gave a structural geometric explanation for it, and it extends the linear combination property from [113] without weight restriction, as well as sharpening several minimum distance bounds from [113]. These are the results that will be presented in this section; they have been published in Adv. Math. Commun. [140]. In Section 2.4, I discuss the LDPC codes from generators and points in Hermitian varieties H(2n + 1; q) with q sufficiently large. This class of codes was studied before in [114], where 1 3=2 for n = 1 the code words of small weight are classified up to weight roughly 2 q as a linear combination of code words of weight 2(q + 1), and for n = 2 it is shown that the only code words c having 0 < wt(c) ≤ 2(q3+q2) have weight 2(q3+1) and 2(q3+q2) and each comes from one specific construction. We extend the second result to arbitrary n and add a classification similar to the first result: the only code words c having 0 < wt(c) ≤ 4q2n−2(q − 1) arise from n different constructions, the minimum distance is in general 2q2n−4(q3 +1) for n ≥ 2; and for every δ > 0 the code words up to weight δq2n−1 are a linear combination of these n smallest types, when q is sufficiently large compared to δ. These results are joint work with M. De Boeck and are accepted for publication in Adv. Math. Commun. [30]. In Section 2.5, I discuss two more infinite families of LDPC codes, derived from the partial ∗ geometries S(K) and T2 (K) with K a maximal arc. For the first construction, I show that swapping the role of points and lines yields an equivalent code, and I extend an earlier result from [23] from hyperovals to Denniston arcs and Mathon arcs. For the second construction, I pose and discuss a conjecture that relates the minimum distance of this code to the existence of certain (q + t; t)-arcs of type (0; 2; t), to which also the next chapter is devoted. I provide a partial proof of this conjecture for the case k = 4. These results were presented at WCC 2011 as ongoing research, but I did not deem them strong enough for journal publication. Chapter 3 is about (q + t; t)-arcs of type (0; 2; t), or shortly KMq;t-arcs, in Desarguesian projective planes of even order. In Section 3.1, I discuss the motivations for studying these arcs, as well as the state of the art. In Section 3.2, I discuss an elegant basis for the projective plane code, and I pose a motivated conjecture, supported by computer simulations, on how linear dependency between incidence vectors of lines translates to the existence of certain KMq;t-arcs. In Section 3.3, I prove this conjecture for the case k = q=2, and thereby indirectly provide an alternative proof for the classification of the projective triads [122, 131]. In Section 3.4 I present my main result on this topic: despite being unable to prove the conjectures posed in Section 3.2, I used them as inspiration to invent a new construction technique for KMq;q=4- arcs, and subsequently proved this new construction by different means. This resulted in both a new infinite class of KMq;t-arcs and a great support for the plausibility of the conjectures posed in Section 3.2. The results in this chapter have been published in Finite Fields Appl. [141]. Chapter 4 discusses optimal blocking multisets, i.e. blocking multisets which m-fold block the hyperplanes of PG(t; q) with as few points as possible. Next to this intrinsic geometric motivation, Section 4.1 discusses another known motivation from coding theory, related to CONTENTS 3 highly divisible Griesmer codes. Section 4.2 discusses a new motivation: when writing the multisets as linear combinations of hyperplanes, it turns out that these are exactly the ones that have all coefficients nonnegative. Armed with this new observation, Section 4.3 improves on almost all known results on this class of multisets. Finally, Section 4.4 demonstrates yet another link with coding theory, namely a link with projective space codes over the ring of integers modulo a prime power.