Best Known (85, 85+20, s)-Nets in Base 7
(85, 85+20, 11766)-Net over F7 — Constructive and digital
Digital (85, 105, 11766)-net over F7, using
- 71 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(7104, 11766, F7, 20, 20) (dual of [(11766, 20), 235216, 21]-NRT-code), using
(85, 85+20, 96099)-Net over F7 — Digital
Digital (85, 105, 96099)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7105, 96099, F7, 20) (dual of [96099, 95994, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(7105, 117664, F7, 20) (dual of [117664, 117559, 21]-code), using
- construction XX applied to Ce(19) ⊂ Ce(17) ⊂ Ce(16) [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(785, 117649, F7, 17) (dual of [117649, 117564, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(71, 14, F7, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), 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(19) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(7105, 117664, F7, 20) (dual of [117664, 117559, 21]-code), using
(85, 85+20, large)-Net in Base 7 — Upper bound on s
There is no (85, 105, large)-net in base 7, because
- 18 times m-reduction [i] would yield (85, 87, large)-net in base 7, but