Best Known (63−10, 63, s)-Nets in Base 7
(63−10, 63, 164723)-Net over F7 — Constructive and digital
Digital (53, 63, 164723)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 13)-net over F7, using
- 7 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (47, 57, 164710)-net over F7, using
- net defined by OOA [i] based on linear OOA(757, 164710, F7, 10, 10) (dual of [(164710, 10), 1647043, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(757, 823550, F7, 10) (dual of [823550, 823493, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(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(9) ⊂ Ce(8) [i] based on
- OA 5-folding and stacking [i] based on linear OA(757, 823550, F7, 10) (dual of [823550, 823493, 11]-code), using
- net defined by OOA [i] based on linear OOA(757, 164710, F7, 10, 10) (dual of [(164710, 10), 1647043, 11]-NRT-code), using
- digital (1, 6, 13)-net over F7, using
(63−10, 63, 823577)-Net over F7 — Digital
Digital (53, 63, 823577)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(763, 823577, F7, 10) (dual of [823577, 823514, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
- linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(729, 823543, F7, 5) (dual of [823543, 823514, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(76, 34, F7, 4) (dual of [34, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- 1 times truncation [i] based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
(63−10, 63, large)-Net in Base 7 — Upper bound on s
There is no (53, 63, large)-net in base 7, because
- 8 times m-reduction [i] would yield (53, 55, large)-net in base 7, but