Best Known (70, 70+12, s)-Nets in Base 7
(70, 70+12, 960803)-Net over F7 — Constructive and digital
Digital (70, 82, 960803)-net over F7, using
- net defined by OOA [i] based on linear OOA(782, 960803, F7, 12, 12) (dual of [(960803, 12), 11529554, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(782, 5764818, F7, 12) (dual of [5764818, 5764736, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(781, 5764801, F7, 12) (dual of [5764801, 5764720, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(765, 5764801, F7, 10) (dual of [5764801, 5764736, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(71, 17, F7, 1) (dual of [17, 16, 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(11) ⊂ Ce(9) [i] based on
- OA 6-folding and stacking [i] based on linear OA(782, 5764818, F7, 12) (dual of [5764818, 5764736, 13]-code), using
(70, 70+12, 5285897)-Net over F7 — Digital
Digital (70, 82, 5285897)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(782, 5285897, F7, 12) (dual of [5285897, 5285815, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(782, 5764818, F7, 12) (dual of [5764818, 5764736, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(781, 5764801, F7, 12) (dual of [5764801, 5764720, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(765, 5764801, F7, 10) (dual of [5764801, 5764736, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(71, 17, F7, 1) (dual of [17, 16, 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(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(782, 5764818, F7, 12) (dual of [5764818, 5764736, 13]-code), using
(70, 70+12, large)-Net in Base 7 — Upper bound on s
There is no (70, 82, large)-net in base 7, because
- 10 times m-reduction [i] would yield (70, 72, large)-net in base 7, but