Optimal Binary Subspace Codes of Length 6, Constant Dimension 3 and Minimum Distance 4

It is shown that the maximum size of a binary subspace code of packet length $v=6$, minimum subspace distance $d=4$, and constant dimension $k=3$ is $M=77$; in Finite Geometry terms, the maximum number of planes in $\operatorname{PG}(5,2)$ mutually intersecting in at most a point is $77$. Optimal binary $(v,M,d;k)=(6,77,4;3)$ subspace codes are classified into $5$ isomorphism types, and a computer-free construction of one isomorphism type is provided. The construction uses both geometry and finite fields theory and generalizes to any $q$, yielding a new family of $q$-ary $(6,q^6+2q^2+2q+1,4;3)$ subspace codes

Johnson type bounds for mixed dimension subspace codes

Subspace codes, i.e., sets of subspaces of $\mathbb{F}_q^v$, are applied in random linear network coding. Here we give improved upper bounds for their cardinalities based on the Johnson bound for constant dimension codes.Comment: 16 pages, typos correcte

Classification of large partial plane spreads in PG(6,2)PG(6,2) and related combinatorial objects

In this article, the partial plane spreads in $PG(6,2)$ of maximum possible size $17$ and of size $16$ are classified. Based on this result, we obtain the classification of the following closely related combinatorial objects: Vector space partitions of $PG(6,2)$ of type $(3^{16} 4^1)$, binary $3\times 4$ MRD codes of minimum rank distance $3$, and subspace codes with parameters $(7,17,6)_2$ and $(7,34,5)_2$.Comment: 31 pages, 9 table

Mini-Collagens in Hydra Nematocytes

We have isolated and characterized four collagen-related c-DNA clones (N-COL 1, N-COL 2, N-COL 3, N-COL 4) that are highly expressed in developing nematocytes in hydra. All four c-DNAs as well as their corresponding transcripts are small in size (600-1,000 bp). The deduced amino acid sequences show that they contain a central region consisting of 14 to 16 Gly-X-Y triplets. This region is flanked amino-terminal by a stretch of 14-23 proline residues and carboxy-terminal by a stretch of 6-9 prolines. At the NH2- and COOH-termini are repeated patterns of cysteine residues that are highly conserved between the molecules. A model is proposed which consists of a central stable collagen triple helix of 12-14 nm length from which three 9-22 nm long polyproline II type helices emerge at both ends. Disulfide linkage between cysteine- rich segments in these helices could lead to the formation of oligomeric network structures. Electrophoretic characterization of nematocyst extracts allows resolution of small proline-rich polypeptides that correspond in size to the cloned sequences