Best Known (65, 65+14, s)-Nets in Base 5
(65, 65+14, 11163)-Net over F5 — Constructive and digital
Digital (65, 79, 11163)-net over F5, using
- net defined by OOA [i] based on linear OOA(579, 11163, F5, 14, 14) (dual of [(11163, 14), 156203, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(579, 78141, F5, 14) (dual of [78141, 78062, 15]-code), using
- construction X4 applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(578, 78125, F5, 14) (dual of [78125, 78047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(564, 78125, F5, 12) (dual of [78125, 78061, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(515, 16, F5, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,5)), using
- dual of repetition code with length 16 [i]
- linear OA(51, 16, F5, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(13) ⊂ Ce(11) [i] based on
- OA 7-folding and stacking [i] based on linear OA(579, 78141, F5, 14) (dual of [78141, 78062, 15]-code), using
(65, 65+14, 46188)-Net over F5 — Digital
Digital (65, 79, 46188)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(579, 46188, F5, 14) (dual of [46188, 46109, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(579, 78140, F5, 14) (dual of [78140, 78061, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(578, 78125, F5, 14) (dual of [78125, 78047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(564, 78125, F5, 12) (dual of [78125, 78061, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(51, 15, F5, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(579, 78140, F5, 14) (dual of [78140, 78061, 15]-code), using
(65, 65+14, large)-Net in Base 5 — Upper bound on s
There is no (65, 79, large)-net in base 5, because
- 12 times m-reduction [i] would yield (65, 67, large)-net in base 5, but