Best Known (42−8, 42, s)-Nets in Base 7
(42−8, 42, 29425)-Net over F7 — Constructive and digital
Digital (34, 42, 29425)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 13)-net over F7, using
- 8 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (29, 37, 29412)-net over F7, using
- net defined by OOA [i] based on linear OOA(737, 29412, F7, 8, 8) (dual of [(29412, 8), 235259, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(737, 117648, F7, 8) (dual of [117648, 117611, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(737, 117649, F7, 8) (dual of [117649, 117612, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(737, 117649, F7, 8) (dual of [117649, 117612, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(737, 117648, F7, 8) (dual of [117648, 117611, 9]-code), using
- net defined by OOA [i] based on linear OOA(737, 29412, F7, 8, 8) (dual of [(29412, 8), 235259, 9]-NRT-code), using
- digital (1, 5, 13)-net over F7, using
(42−8, 42, 117673)-Net over F7 — Digital
Digital (34, 42, 117673)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(742, 117673, F7, 8) (dual of [117673, 117631, 9]-code), using
- construction XX applied to Ce(7) ⊂ Ce(3) ⊂ Ce(2) [i] based on
- linear OA(737, 117649, F7, 8) (dual of [117649, 117612, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(719, 117649, F7, 4) (dual of [117649, 117630, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(713, 117649, F7, 3) (dual of [117649, 117636, 4]-code or 117649-cap in PG(12,7)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(74, 23, F7, 3) (dual of [23, 19, 4]-code or 23-cap in PG(3,7)), 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(7) ⊂ Ce(3) ⊂ Ce(2) [i] based on
(42−8, 42, large)-Net in Base 7 — Upper bound on s
There is no (34, 42, large)-net in base 7, because
- 6 times m-reduction [i] would yield (34, 36, large)-net in base 7, but