Best Known (109−21, 109, s)-Nets in Base 7
(109−21, 109, 11765)-Net over F7 — Constructive and digital
Digital (88, 109, 11765)-net over F7, using
- net defined by OOA [i] based on linear OOA(7109, 11765, F7, 21, 21) (dual of [(11765, 21), 246956, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(7109, 117651, F7, 21) (dual of [117651, 117542, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(7109, 117655, F7, 21) (dual of [117655, 117546, 22]-code), using
- 1 times truncation [i] based on linear OA(7110, 117656, F7, 22) (dual of [117656, 117546, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [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]
- 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(71, 7, F7, 1) (dual of [7, 6, 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(21) ⊂ Ce(19) [i] based on
- 1 times truncation [i] based on linear OA(7110, 117656, F7, 22) (dual of [117656, 117546, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(7109, 117655, F7, 21) (dual of [117655, 117546, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(7109, 117651, F7, 21) (dual of [117651, 117542, 22]-code), using
(109−21, 109, 84085)-Net over F7 — Digital
Digital (88, 109, 84085)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7109, 84085, F7, 21) (dual of [84085, 83976, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(7109, 117650, F7, 21) (dual of [117650, 117541, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 117650 | 712−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(7109, 117650, F7, 21) (dual of [117650, 117541, 22]-code), using
(109−21, 109, large)-Net in Base 7 — Upper bound on s
There is no (88, 109, large)-net in base 7, because
- 19 times m-reduction [i] would yield (88, 90, large)-net in base 7, but