Best Known (55−10, 55, s)-Nets in Base 7
(55−10, 55, 23544)-Net over F7 — Constructive and digital
Digital (45, 55, 23544)-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 (39, 49, 23531)-net over F7, using
- net defined by OOA [i] based on linear OOA(749, 23531, F7, 10, 10) (dual of [(23531, 10), 235261, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(749, 117655, F7, 10) (dual of [117655, 117606, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(749, 117649, F7, 10) (dual of [117649, 117600, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(743, 117649, F7, 9) (dual of [117649, 117606, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(70, 6, F7, 0) (dual of [6, 6, 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(749, 117655, F7, 10) (dual of [117655, 117606, 11]-code), using
- net defined by OOA [i] based on linear OOA(749, 23531, F7, 10, 10) (dual of [(23531, 10), 235261, 11]-NRT-code), using
- digital (1, 6, 13)-net over F7, using
(55−10, 55, 117679)-Net over F7 — Digital
Digital (45, 55, 117679)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(755, 117679, F7, 10) (dual of [117679, 117624, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
- linear OA(749, 117649, F7, 10) (dual of [117649, 117600, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(725, 117649, F7, 5) (dual of [117649, 117624, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(76, 30, F7, 4) (dual of [30, 24, 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
(55−10, 55, large)-Net in Base 7 — Upper bound on s
There is no (45, 55, large)-net in base 7, because
- 8 times m-reduction [i] would yield (45, 47, large)-net in base 7, but