Best Known (87, 87+21, s)-Nets in Base 7
(87, 87+21, 11764)-Net over F7 — Constructive and digital
Digital (87, 108, 11764)-net over F7, using
- net defined by OOA [i] based on linear OOA(7108, 11764, F7, 21, 21) (dual of [(11764, 21), 246936, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(7108, 117641, F7, 21) (dual of [117641, 117533, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(7108, 117648, F7, 21) (dual of [117648, 117540, 22]-code), using
- 1 times truncation [i] based on linear OA(7109, 117649, F7, 22) (dual of [117649, 117540, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 1 times truncation [i] based on linear OA(7109, 117649, F7, 22) (dual of [117649, 117540, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(7108, 117648, F7, 21) (dual of [117648, 117540, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(7108, 117641, F7, 21) (dual of [117641, 117533, 22]-code), using
(87, 87+21, 75899)-Net over F7 — Digital
Digital (87, 108, 75899)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7108, 75899, F7, 21) (dual of [75899, 75791, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(7108, 117648, F7, 21) (dual of [117648, 117540, 22]-code), using
- 1 times truncation [i] based on linear OA(7109, 117649, F7, 22) (dual of [117649, 117540, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 1 times truncation [i] based on linear OA(7109, 117649, F7, 22) (dual of [117649, 117540, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(7108, 117648, F7, 21) (dual of [117648, 117540, 22]-code), using
(87, 87+21, large)-Net in Base 7 — Upper bound on s
There is no (87, 108, large)-net in base 7, because
- 19 times m-reduction [i] would yield (87, 89, large)-net in base 7, but