Best Known (52, 52+10, s)-Nets in Base 7
(52, 52+10, 164718)-Net over F7 — Constructive and digital
Digital (52, 62, 164718)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 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 (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 (0, 5, 8)-net over F7, using
(52, 52+10, 823570)-Net over F7 — Digital
Digital (52, 62, 823570)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(762, 823570, F7, 10) (dual of [823570, 823508, 11]-code), using
- construction XX applied to Ce(9) ⊂ Ce(5) ⊂ 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(736, 823543, F7, 6) (dual of [823543, 823507, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(74, 26, F7, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,7)), using
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(9) ⊂ Ce(5) ⊂ Ce(4) [i] based on
(52, 52+10, large)-Net in Base 7 — Upper bound on s
There is no (52, 62, large)-net in base 7, because
- 8 times m-reduction [i] would yield (52, 54, large)-net in base 7, but