Best Known (65, 77, s)-Nets in Base 7
(65, 77, 137266)-Net over F7 — Constructive and digital
Digital (65, 77, 137266)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- digital (59, 71, 137258)-net over F7, using
- net defined by OOA [i] based on linear OOA(771, 137258, F7, 12, 12) (dual of [(137258, 12), 1647025, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(771, 823548, F7, 12) (dual of [823548, 823477, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(771, 823550, F7, 12) (dual of [823550, 823479, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(771, 823543, F7, 12) (dual of [823543, 823472, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(764, 823543, F7, 11) (dual of [823543, 823479, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 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
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(771, 823550, F7, 12) (dual of [823550, 823479, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(771, 823548, F7, 12) (dual of [823548, 823477, 13]-code), using
- net defined by OOA [i] based on linear OOA(771, 137258, F7, 12, 12) (dual of [(137258, 12), 1647025, 13]-NRT-code), using
- digital (0, 6, 8)-net over F7, using
(65, 77, 823579)-Net over F7 — Digital
Digital (65, 77, 823579)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(777, 823579, F7, 12) (dual of [823579, 823502, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(775, 823575, F7, 12) (dual of [823575, 823500, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(771, 823543, F7, 12) (dual of [823543, 823472, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(11) ⊂ Ce(7) [i] based on
- linear OA(775, 823577, F7, 11) (dual of [823577, 823502, 12]-code), using Gilbert–Varšamov bound and bm = 775 > Vbs−1(k−1) = 2 392012 488029 206879 658599 954080 943155 411035 176907 714533 502017 [i]
- linear OA(70, 2, F7, 0) (dual of [2, 2, 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(775, 823575, F7, 12) (dual of [823575, 823500, 13]-code), using
- construction X with Varšamov bound [i] based on
(65, 77, large)-Net in Base 7 — Upper bound on s
There is no (65, 77, large)-net in base 7, because
- 10 times m-reduction [i] would yield (65, 67, large)-net in base 7, but