We introduce a linear programming method to obtain bounds on the cardinality of codes in Grassmannian spaces for the chordal distance. We obtain explicit bounds, and an asymptotic bound that improves on the Hamming bound. Our approach generalizes the approach originally developed by P. Delsarte and Kabatianski-Levenshtein for compact two-point homogeneous spaces.
Equidistant codes over vector spaces are considered. For $k$-dimensional subspaces over a large vector space the largest code is always a sunflower. We present several simple constructions for such codes which might produce the largest non-sunflower codes. A novel construction, based on the Plücker embedding, for 1-intersecting codes of $k$-dimensional subspaces over $\F_q^n$, $n \geq \binom{k+1}{2}$, where the code size is $\frac{q^{k+1}-1}{q-1}$ is presented. Finally, we present a related construction which generates equidistant constant rank codes with matrices of size $n \times \binom{n}{2…
We present a concise description of Orthogonal Polar Grassmann Codes and motivate their relevance. We also describe efficient encoding and decoding algorithms for the case of Line Grassmannians and introduce some open problems.
A $k$-polar Grassmannian is the geometry having as pointset the set of all $k$-dimensional subspaces of a vector space $V$ which are totally isotropic for a given non-degenerate bilinear form $μ$ defined on $V.$ Hence it can be regarded as a subgeometry of the ordinary $k$-Grassmannian. In this paper we deal with orthogonal line Grassmannians and with symplectic line Grassmannians, i.e. we assume $k=2$ and $μ$ a non-degenerate symmetric or alternating form. We will provide a method to efficiently enumerate the pointsets of both orthogonal and symplectic line Grassmannians. This has several nic…
Polar Grassmann codes of orthogonal type have been introduced in I. Cardinali and L. Giuzzi, \emph{Codes and caps from orthogonal Grassmannians}, {Finite Fields Appl.} {\bf 24} (2013), 148-169. They are subcodes of the Grassmann code arising from the projective system defined by the Plücker embedding of a polar Grassmannian of orthogonal type. In the present paper we fully determine the minimum distance of line polar Grassmann Codes of orthogonal type for $q$ odd.
We introduce the Symplectic Grassmann codes as projective codes defined by symplectic Grassmannians, in analogy with the orthogonal Grassmann codes introduced in [4]. Note that the Lagrangian-Grassmannian codes are a special class of Symplectic Grassmann codes. We describe the weight enumerator of the Lagrangian--Grassmannian codes of rank $2$ and $3$ and we determine the minimum distance of the line Symplectic Grassmann codes.
An equiangular tight frame (ETF) is a sequence of vectors in a Hilbert space that achieves equality in the Welch bound and so has minimal coherence. More generally, an equichordal tight fusion frame (ECTFF) is a sequence of equi-dimensional subspaces of a Hilbert space that achieves equality in Conway, Hardin and Sloane's simplex bound. Every ECTFF is a type of optimal Grassmannian code, that is, an optimal packing of equi-dimensional subspaces of a Hilbert space. We construct ECTFFs by exploiting new relationships between known ETFs. Harmonic ETFs equate to difference sets for finite abelian …
A complex spherical code is a finite subset on the unit sphere in $\mathbb{C}^d$. A fundamental problem on complex spherical codes is to find upper bounds for those with prescribed inner products. In this paper, we determine the irreducible decomposition under the action of the one-point stabilizer of the unitary group $U(d)$ on the polynomial ring $\mathbb{C}[z_1\ldots,z_d,\bar{z}_1,\ldots,\bar{z}_d]$ in order to obtain the semidefinite programming bounds for complex spherical 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 $n$.
The Grassmannian $\mathcal{G}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space $\mathbb{F}_q^n$. Kötter and Kschischang showed that codes in Grassmannian space can be used for error-correction in random network coding. On the other hand, these codes are $q$-analogs of codes in the Johnson scheme, i.e., constant dimension codes. These codes of the Grassmannian $\mathcal{G}_q(n,k)$ also form a family of $q$-analogs of block designs and they are called subspace designs. In this paper, we examine one of the last families of $q$-analogs of block designs which was not consider…
For $n,d,w \in \mathbb{N}$, let $A(n,d,w)$ denote the maximum size of a binary code of word length $n$, minimum distance $d$ and constant weight $w$. Schrijver recently showed using semidefinite programming that $A(23,8,11)=1288$, and the second author that $A(22,8,11)=672$ and $A(22,8,10)=616$. Here we show uniqueness of the codes achieving these bounds. Let $A(n,d)$ denote the maximum size of a binary code of word length $n$ and minimum distance $d$. Gijswijt, Mittelmann and Schrijver showed that $A(20,8)=256$. We show that there are several nonisomorphic codes achieving this bound, and clas…
We give one more proof of the first linear programming bound for binary codes, following the line of work initiated by Friedman and Tillich. The new argument is somewhat similar to previous proofs, but we believe it to be both simpler and more intuitive. Moreover, it provides the following 'geometric' explanation for the bound. A binary code with minimal distance $δn$ is small because the projections of the characteristic functions of its elements on the subspace spanned by the Walsh-Fourier characters of weight up to $\left(\frac 12 - \sqrt{δ(1-δ)}\right) \cdot n$ are essentially independent.…
For nonnegative integers $n,d,w$, let $A(n,d,w)$ be the maximum size of a code $C \subseteq \mathbb{F}_2^n$ with constant weight $w$ and minimum distance at least $d$. We consider two semidefinite programs based on quadruples of code words that yield several new upper bounds on $A(n,d,w)$. The new upper bounds imply that $A(22,8,10)=616$ and $A(22,8,11)=672$. Lower bounds on $A(22,8,10)$ and $A(22,8,11)$ are obtained from the $(n,d)=(22,7)$ shortened Golay code of size $2048$. It can be concluded that the shortened Golay code is a union of constant weight $w$ codes of sizes $A(22,8,w)$.
