Maximal partial spreads and the modular n-queen problem III

AbstractMaximal partial spreads in PG(3,q)q=pk,p odd prime and q⩾7, are constructed for any integer n in the interval (q2+1)/2+6⩽n⩽(5q2+4q−1)/8 in the case q+1≡0,±2,±4,±6,±10,12(mod24). In all these cases, maximal partial spreads of the size (q2+1)/2+n have also been constructed for some small values of the integer n. These values depend on q and are mainly n=3 and n=4. Combining these results with previous results of the author and with that of others we can conclude that there exist maximal partial spreads in PG(3,q),q=pk where p is an odd prime and q⩾7, of size n for any integer n in the interval (q2+1)/2+6⩽n⩽q2−q+2

On perfect 1-mathcalEmathcal E-error-correcting codes

We generalize the concept of perfect Lee-error-correcting codes, and present constructions of this new class of perfect codes that are called perfect 1-$mathcal{E}$-error-correcting codes. We also show that in some cases such codes contain quite a few perfect 1-error-correcting $q$-ary Hamming codes as subsets

Some necessary conditions for vector space partitions

Some new necessary conditions for the existence of vector space partitions are derived. They are applied to the problem of finding the maximum number of spaces of dimension t in a vector space partition of V(2t,q) that contains m_d spaces of dimension d, where t/2<d<t, and also spaces of other dimensions. It is also discussed how this problem is related to maximal partial t-spreads in V(2t,q). We also give a lower bound for the number of spaces in a vector space partition and verify that this bound is tight.Comment: 19 pages; corrected typos and rewritten introductio

Perfect 1-error-correcting Lipschitz weight codes

Let $pi$ be a Lipschitz prime and $p=pipi^star$. Perfect 1-error-correcting codes in $H(mathbb{Z})_pi^n$ are constructed for every prime number $pequiv1(bmod;4)$. This completes a result of the authors in an earlier work, emph{Perfect Mannheim, Lipschitz and Hurwitz weight codes}, (Mathematical Communications, Vol 19, No 2, pp. 253 -- 276 (2014)), where a construction is given in the case $pequiv3,(bmod;4)$

Perfect Mannheim, Lipschitz and Hurwitz weight codes

The set of residue classes modulo an element in the rings of Gaussian integers,Lipschitz integers and Hurwitz integers, respectively, is used as alphabets to form the words of error correcting codes. An error occurs as the addition of an element in a set E to the letter in one of the positions of a word. If E is a group of units in the original rings, then we obtain the Mannheim, Lipschitz and Hurwitz metrics, respectively. Some new perfect 1-error-correcting codes in these metrics are constructed. The existence of perfect 2-error-correcting codes is investigated by computer search

On the structure of non-full-rank perfect codes

The Krotov combining construction of perfect 1-error-correcting binary codes from 2000 and a theorem of Heden saying that every non-full-rank perfect 1-error-correcting binary code can be constructed by this combining construction is generalized to the $q$-ary case. Simply, every non-full-rank perfect code $C$ is the union of a well-defined family of $\mu$-components $K_\mu$, where $\mu$ belongs to an "outer" perfect code $C^*$, and these components are at distance three from each other. Components from distinct codes can thus freely be combined to obtain new perfect codes. The Phelps general product construction of perfect binary code from 1984 is generalized to obtain $\mu$-components, and new lower bounds on the number of perfect 1-error-correcting $q$-ary codes are presented.Comment: 8 page