New and Updated Semidefinite Programming Bounds for Subspace Codes

We show that $A_2(7,4) \leq 388$ and, more generally, $A_q(7,4) \leq (q^2-q+1)[7]_q + q^4 - 2q^3 + 3q^2 - 4q + 4$ by semidefinite programming for $q \leq 101$. Furthermore, we extend results by Bachoc et al. on SDP bounds for $A_2(n,d)$, where $d$ is odd and $n$ is small, to $A_q(n,d)$ for small $q$ and small $n$

Tables of subspace codes

One of the main problems of subspace coding asks for the maximum possible cardinality of a subspace code with minimum distance at least $d$ over $\mathbb{F}_q^n$, where the dimensions of the codewords, which are vector spaces, are contained in $K\subseteq\{0,1,\dots,n\}$. In the special case of $K=\{k\}$ one speaks of constant dimension codes. Since this (still) emerging field is very prosperous on the one hand side and there are a lot of connections to classical objects from Galois geometry it is a bit difficult to keep or to obtain an overview about the current state of knowledge. To this end we have implemented an on-line database of the (at least to us) known results at \url{subspacecodes.uni-bayreuth.de}. The aim of this recurrently updated technical report is to provide a user guide how this technical tool can be used in research projects and to describe the so far implemented theoretic and algorithmic knowledge.Comment: 44 pages, 6 tables, 7 screenshot

A new upper bound for subspace codes

It is shown that the maximum size $A_2(8,6;4)$ of a binary subspace code of packet length $v=8$, minimum subspace distance $d=4$, and constant dimension $k=4$ is at most $272$. In Finite Geometry terms, the maximum number of solids in $\operatorname{PG}(7,2)$, mutually intersecting in at most a point, is at most $272$. Previously, the best known upper bound $A_2(8,6;4)\le 289$ was implied by the Johnson bound and the maximum size $A_2(7,6;3)=17$ of partial plane spreads in $\operatorname{PG}(6,2)$. The result was obtained by combining the classification of subspace codes with parameters $(7,17,6;3)_2$ and $(7,34,5;\{3,4\})_2$ with integer linear programming techniques. The classification of $(7,33,5;\{3,4\})_2$ subspace codes is obtained as a byproduct.Comment: 9 page

I\u27m Polynesian Too: Philosophy of Assimilation, Cosmopolitanism, Colonialism, Race, & Culture

Finding identity is difficult for mixed race and culture Polynesian Americans because there is no full integration into either racial/cultural side. For many Polynesian Americans (mixed race or not), finding an ethnic, cultural, and philosophical identity is a life-long struggle that constantly toils in matters tied to their souls and well being: issues of right and wrong, gender roles, morals/ethics, acceptance, and what it means to be human. For Polynesians and mixed race Polynesians, tribulation and alienation stem from the assimilation model that is present in the world today. “American Consumerist Cosmopolitanism,” as descended from colonialism, has impacted the well-being of Polynesian Americans (mixed race or not) for the worse. I will argue that the values of Polynesian culture are best preserved by a reevaluation of racial categories and ethnic practices in light of the unique colonialist history of Polynesians and that we need to move toward a model of Pluralistic Cosmopolitanism, which promotes true multicultural autonomy and both inter- and intra-cultural acceptance, rather than elitism. To explain and back this, I give brief histories of the Samoan and Hawaiian people, as well as some background in Polynesian philosophy, relevant sociological issues, assimilation/acculturation models, and look at racial philosophy, particularly in how these issues impact the continuation of Samoan and Hawaiian culture

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

Generalized vector space partitions

A vector space partition $\mathcal{P}$ in $\mathbb{F}_q^v$ is a set of subspaces such that every $1$-dimensional subspace of $\mathbb{F}_q^v$ is contained in exactly one element of $\mathcal{P}$. Replacing "every point" by "every $t$-dimensional subspace", we generalize this notion to vector space $t$-partitions and study their properties. There is a close connection to subspace codes and some problems are even interesting and unsolved for the set case $q=1$.Comment: 12 pages, typos correcte