Best Known (31−8, 31, s)-Nets in Base 7
(31−8, 31, 4201)-Net over F7 — Constructive and digital
Digital (23, 31, 4201)-net over F7, using
- net defined by OOA [i] based on linear OOA(731, 4201, F7, 8, 8) (dual of [(4201, 8), 33577, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(731, 16804, F7, 8) (dual of [16804, 16773, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(731, 16804, F7, 8) (dual of [16804, 16773, 9]-code), using
(31−8, 31, 8403)-Net over F7 — Digital
Digital (23, 31, 8403)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(731, 8403, F7, 2, 8) (dual of [(8403, 2), 16775, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(731, 16806, F7, 8) (dual of [16806, 16775, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using
- OOA 2-folding [i] based on linear OA(731, 16806, F7, 8) (dual of [16806, 16775, 9]-code), using
(31−8, 31, 1307406)-Net in Base 7 — Upper bound on s
There is no (23, 31, 1307407)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 157 775607 949600 035224 760553 > 731 [i]