Best Known (73−14, 73, s)-Nets in Base 7
(73−14, 73, 16807)-Net over F7 — Constructive and digital
Digital (59, 73, 16807)-net over F7, using
- t-expansion [i] based on digital (58, 73, 16807)-net over F7, using
- net defined by OOA [i] based on linear OOA(773, 16807, F7, 15, 15) (dual of [(16807, 15), 252032, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [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]
- OOA 7-folding and stacking with additional row [i] based on linear OA(773, 117650, F7, 15) (dual of [117650, 117577, 16]-code), using
- net defined by OOA [i] based on linear OOA(773, 16807, F7, 15, 15) (dual of [(16807, 15), 252032, 16]-NRT-code), using
(73−14, 73, 103698)-Net over F7 — Digital
Digital (59, 73, 103698)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(773, 103698, F7, 14) (dual of [103698, 103625, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(773, 117655, F7, 14) (dual of [117655, 117582, 15]-code), using
- 1 times truncation [i] based on linear OA(774, 117656, F7, 15) (dual of [117656, 117582, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(773, 117649, F7, 15) (dual of [117649, 117576, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(767, 117649, F7, 13) (dual of [117649, 117582, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(71, 7, F7, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- 1 times truncation [i] based on linear OA(774, 117656, F7, 15) (dual of [117656, 117582, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(773, 117655, F7, 14) (dual of [117655, 117582, 15]-code), using
(73−14, 73, large)-Net in Base 7 — Upper bound on s
There is no (59, 73, large)-net in base 7, because
- 12 times m-reduction [i] would yield (59, 61, large)-net in base 7, but