Best Known (78−23, 78, s)-Nets in Base 7
(78−23, 78, 218)-Net over F7 — Constructive and digital
Digital (55, 78, 218)-net over F7, using
- 71 times duplication [i] based on digital (54, 77, 218)-net over F7, using
- net defined by OOA [i] based on linear OOA(777, 218, F7, 23, 23) (dual of [(218, 23), 4937, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(777, 2399, F7, 23) (dual of [2399, 2322, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(777, 2401, F7, 23) (dual of [2401, 2324, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(777, 2401, F7, 23) (dual of [2401, 2324, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(777, 2399, F7, 23) (dual of [2399, 2322, 24]-code), using
- net defined by OOA [i] based on linear OOA(777, 218, F7, 23, 23) (dual of [(218, 23), 4937, 24]-NRT-code), using
(78−23, 78, 1803)-Net over F7 — Digital
Digital (55, 78, 1803)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(778, 1803, F7, 23) (dual of [1803, 1725, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(778, 2406, F7, 23) (dual of [2406, 2328, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(777, 2405, F7, 23) (dual of [2405, 2328, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(777, 2401, F7, 23) (dual of [2401, 2324, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(773, 2401, F7, 22) (dual of [2401, 2328, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(777, 2405, F7, 23) (dual of [2405, 2328, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(778, 2406, F7, 23) (dual of [2406, 2328, 24]-code), using
(78−23, 78, 673821)-Net in Base 7 — Upper bound on s
There is no (55, 78, 673822)-net in base 7, because
- 1 times m-reduction [i] would yield (55, 77, 673822)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 118182 598039 582389 345608 332188 887158 942222 962259 154703 337154 231353 > 777 [i]