Best Known (47, 62, s)-Nets in Base 7
(47, 62, 2401)-Net over F7 — Constructive and digital
Digital (47, 62, 2401)-net over F7, using
- 71 times duplication [i] based on digital (46, 61, 2401)-net over F7, using
- net defined by OOA [i] based on linear OOA(761, 2401, F7, 15, 15) (dual of [(2401, 15), 35954, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(761, 16808, F7, 15) (dual of [16808, 16747, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, 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(761, 16808, F7, 15) (dual of [16808, 16747, 16]-code), using
- net defined by OOA [i] based on linear OOA(761, 2401, F7, 15, 15) (dual of [(2401, 15), 35954, 16]-NRT-code), using
(47, 62, 8716)-Net over F7 — Digital
Digital (47, 62, 8716)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(762, 8716, F7, 15) (dual of [8716, 8654, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(762, 16813, F7, 15) (dual of [16813, 16751, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(761, 16807, F7, 15) (dual of [16807, 16746, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(756, 16807, F7, 13) (dual of [16807, 16751, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(71, 6, F7, 1) (dual of [6, 5, 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
- discarding factors / shortening the dual code based on linear OA(762, 16813, F7, 15) (dual of [16813, 16751, 16]-code), using
(47, 62, large)-Net in Base 7 — Upper bound on s
There is no (47, 62, large)-net in base 7, because
- 13 times m-reduction [i] would yield (47, 49, large)-net in base 7, but