Best Known (52−26, 52, s)-Nets in Base 81
(52−26, 52, 505)-Net over F81 — Constructive and digital
Digital (26, 52, 505)-net over F81, using
- net defined by OOA [i] based on linear OOA(8152, 505, F81, 26, 26) (dual of [(505, 26), 13078, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(8152, 6565, F81, 26) (dual of [6565, 6513, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8152, 6566, F81, 26) (dual of [6566, 6514, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- linear OA(8151, 6561, F81, 26) (dual of [6561, 6510, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(8147, 6561, F81, 24) (dual of [6561, 6514, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(8152, 6566, F81, 26) (dual of [6566, 6514, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(8152, 6565, F81, 26) (dual of [6565, 6513, 27]-code), using
(52−26, 52, 2006)-Net over F81 — Digital
Digital (26, 52, 2006)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8152, 2006, F81, 3, 26) (dual of [(2006, 3), 5966, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8152, 2188, F81, 3, 26) (dual of [(2188, 3), 6512, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8152, 6564, F81, 26) (dual of [6564, 6512, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8152, 6566, F81, 26) (dual of [6566, 6514, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- linear OA(8151, 6561, F81, 26) (dual of [6561, 6510, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(8147, 6561, F81, 24) (dual of [6561, 6514, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(8152, 6566, F81, 26) (dual of [6566, 6514, 27]-code), using
- OOA 3-folding [i] based on linear OA(8152, 6564, F81, 26) (dual of [6564, 6512, 27]-code), using
- discarding factors / shortening the dual code based on linear OOA(8152, 2188, F81, 3, 26) (dual of [(2188, 3), 6512, 27]-NRT-code), using
(52−26, 52, 3049687)-Net in Base 81 — Upper bound on s
There is no (26, 52, 3049688)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 1742 694515 877524 253228 117813 888694 439612 241106 314256 734822 005699 062287 615470 774729 517945 174657 422721 > 8152 [i]