Best Known (109−20, 109, s)-Nets in Base 7
(109−20, 109, 11768)-Net over F7 — Constructive and digital
Digital (89, 109, 11768)-net over F7, using
- net defined by OOA [i] based on linear OOA(7109, 11768, F7, 20, 20) (dual of [(11768, 20), 235251, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(7109, 117680, F7, 20) (dual of [117680, 117571, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(7109, 117685, F7, 20) (dual of [117685, 117576, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(7103, 117649, F7, 20) (dual of [117649, 117546, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(773, 117649, F7, 15) (dual of [117649, 117576, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(76, 36, F7, 4) (dual of [36, 30, 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(19) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(7109, 117685, F7, 20) (dual of [117685, 117576, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(7109, 117680, F7, 20) (dual of [117680, 117571, 21]-code), using
(109−20, 109, 117685)-Net over F7 — Digital
Digital (89, 109, 117685)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7109, 117685, F7, 20) (dual of [117685, 117576, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(7103, 117649, F7, 20) (dual of [117649, 117546, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(773, 117649, F7, 15) (dual of [117649, 117576, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(76, 36, F7, 4) (dual of [36, 30, 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(19) ⊂ Ce(14) [i] based on
(109−20, 109, large)-Net in Base 7 — Upper bound on s
There is no (89, 109, large)-net in base 7, because
- 18 times m-reduction [i] would yield (89, 91, large)-net in base 7, but