Best Known (70, 70+13, s)-Nets in Base 7
(70, 70+13, 137262)-Net over F7 — Constructive and digital
Digital (70, 83, 137262)-net over F7, using
- 71 times duplication [i] based on digital (69, 82, 137262)-net over F7, using
- net defined by OOA [i] based on linear OOA(782, 137262, F7, 13, 13) (dual of [(137262, 13), 1784324, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(782, 823573, F7, 13) (dual of [823573, 823491, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(782, 823575, F7, 13) (dual of [823575, 823493, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(778, 823543, F7, 13) (dual of [823543, 823465, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(750, 823543, F7, 9) (dual of [823543, 823493, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(74, 32, F7, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,7)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(782, 823575, F7, 13) (dual of [823575, 823493, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(782, 823573, F7, 13) (dual of [823573, 823491, 14]-code), using
- net defined by OOA [i] based on linear OOA(782, 137262, F7, 13, 13) (dual of [(137262, 13), 1784324, 14]-NRT-code), using
(70, 70+13, 823577)-Net over F7 — Digital
Digital (70, 83, 823577)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(783, 823577, F7, 13) (dual of [823577, 823494, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(782, 823575, F7, 13) (dual of [823575, 823493, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(778, 823543, F7, 13) (dual of [823543, 823465, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(750, 823543, F7, 9) (dual of [823543, 823493, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(74, 32, F7, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,7)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(782, 823576, F7, 12) (dual of [823576, 823494, 13]-code), using Gilbert–Varšamov bound and bm = 782 > Vbs−1(k−1) = 1 074520 496494 241139 277116 848866 162544 372433 609779 915809 701206 519671 [i]
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(782, 823575, F7, 13) (dual of [823575, 823493, 14]-code), using
- construction X with Varšamov bound [i] based on
(70, 70+13, large)-Net in Base 7 — Upper bound on s
There is no (70, 83, large)-net in base 7, because
- 11 times m-reduction [i] would yield (70, 72, large)-net in base 7, but