Binary Codes and Caps
نویسندگان
چکیده
The connection between maximal caps (sometimes called complete caps) and certain binary codes called quasi-perfect codes is described. We provide a geometric approach to the foundational work of Davydov and Tombak who have obtained the exact possible sizes of large maximal caps. A new self-contained proof of the existence and the structure of the largest maximal nonaffine cap in PG(n, 2) is given. Combinatorial and geometric consequences are briefly sketched. Some of these, such as the connection with families of symmetric-difference free subsets of a finite set will be developed elsewhere. c © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 275–284, 1998
منابع مشابه
Constructions of Small Complete Caps in Binary Projective Spaces
In the binary projective spaces PG(n,2) k-caps are called large if k > 2n−1 and small if k ≤ 2n−1. In this paper we propose new constructions producing infinite families of small binary complete caps.
متن کاملLong Binary Linear Codes and Large Caps in Projective Space
We obtain, in principle, a complete classification of all long inextendable binary linear codes. Several related constructions and results are presented.
متن کاملThe structure of quaternary quantum caps
We give a geometric description of binary quantum stabilizer codes. In the case of distance d = 4 this leads to the notion of a quaternary quantum cap. We describe several recursive constructions for quantum caps, determine the quantum caps in PG(3, 4) and the cardinalities of quantum caps in PG(4, 4). ∗research partially supported NSA grant H98230-10-1-0159 †The research of this author takes p...
متن کامل7 M ay 2 00 9 New quantum caps in PG ( 4 , 4
Calderbank, Rains, Shor and Sloane (see [6]) showed that error-correction is possible in the context of quantum computations. Quantum stabilizer codes are a class of additive quaternary codes in binary projective spaces, which are self-orthogonal with respect to the symplectic form. A geometric description is given in [5], where also the notion of quantum cap is introduced. Quantum caps corresp...
متن کاملNew quantum caps in PG ( 4 , 4 )
Calderbank, Rains, Shor and Sloane (see [9]) showed that error-correction is possible in the context of quantum computations. Quantum stabilizer codes are a class of additive quaternary codes in binary projective spaces, which are self-orthogonal with respect to the symplectic form. A geometric description is given in [6], where also the notion of quantum cap is introduced. Quantum caps corresp...
متن کامل