Best Known (104−20, 104, s)-Nets in Base 7
(104−20, 104, 11766)-Net over F7 — Constructive and digital
Digital (84, 104, 11766)-net over F7, using
- net defined by OOA [i] based on linear OOA(7104, 11766, F7, 20, 20) (dual of [(11766, 20), 235216, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(7104, 117660, F7, 20) (dual of [117660, 117556, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(7104, 117662, F7, 20) (dual of [117662, 117558, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [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(791, 117649, F7, 18) (dual of [117649, 117558, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(71, 13, F7, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(7104, 117662, F7, 20) (dual of [117662, 117558, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(7104, 117660, F7, 20) (dual of [117660, 117556, 21]-code), using
(104−20, 104, 86251)-Net over F7 — Digital
Digital (84, 104, 86251)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7104, 86251, F7, 20) (dual of [86251, 86147, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(7104, 117662, F7, 20) (dual of [117662, 117558, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [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(791, 117649, F7, 18) (dual of [117649, 117558, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(71, 13, F7, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(7104, 117662, F7, 20) (dual of [117662, 117558, 21]-code), using
(104−20, 104, large)-Net in Base 7 — Upper bound on s
There is no (84, 104, large)-net in base 7, because
- 18 times m-reduction [i] would yield (84, 86, large)-net in base 7, but