Best Known (58, 58+14, s)-Nets in Base 7
(58, 58+14, 16807)-Net over F7 — Constructive and digital
Digital (58, 72, 16807)-net over F7, using
- net defined by OOA [i] based on linear OOA(772, 16807, F7, 14, 14) (dual of [(16807, 14), 235226, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(772, 117649, F7, 14) (dual of [117649, 117577, 15]-code), using
- 1 times truncation [i] based on linear OA(773, 117650, F7, 15) (dual of [117650, 117577, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 117650 | 712−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(773, 117650, F7, 15) (dual of [117650, 117577, 16]-code), using
- OA 7-folding and stacking [i] based on linear OA(772, 117649, F7, 14) (dual of [117649, 117577, 15]-code), using
(58, 58+14, 88174)-Net over F7 — Digital
Digital (58, 72, 88174)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(772, 88174, F7, 14) (dual of [88174, 88102, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(772, 117649, F7, 14) (dual of [117649, 117577, 15]-code), using
- 1 times truncation [i] based on linear OA(773, 117650, F7, 15) (dual of [117650, 117577, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 117650 | 712−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(773, 117650, F7, 15) (dual of [117650, 117577, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(772, 117649, F7, 14) (dual of [117649, 117577, 15]-code), using
(58, 58+14, large)-Net in Base 7 — Upper bound on s
There is no (58, 72, large)-net in base 7, because
- 12 times m-reduction [i] would yield (58, 60, large)-net in base 7, but