Best Known (46, 54, s)-Nets in Base 7
(46, 54, 1441213)-Net over F7 — Constructive and digital
Digital (46, 54, 1441213)-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 (41, 49, 1441200)-net over F7, using
- net defined by OOA [i] based on linear OOA(749, 1441200, F7, 8, 8) (dual of [(1441200, 8), 11529551, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(749, 5764800, F7, 8) (dual of [5764800, 5764751, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(749, 5764800, F7, 8) (dual of [5764800, 5764751, 9]-code), using
- net defined by OOA [i] based on linear OOA(749, 1441200, F7, 8, 8) (dual of [(1441200, 8), 11529551, 9]-NRT-code), using
- digital (1, 5, 13)-net over F7, using
(46, 54, 5764831)-Net over F7 — Digital
Digital (46, 54, 5764831)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(754, 5764831, F7, 8) (dual of [5764831, 5764777, 9]-code), using
- construction XX applied to Ce(7) ⊂ Ce(3) ⊂ Ce(2) [i] based on
- linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(725, 5764801, F7, 4) (dual of [5764801, 5764776, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(717, 5764801, F7, 3) (dual of [5764801, 5764784, 4]-code or 5764801-cap in PG(16,7)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(74, 29, F7, 3) (dual of [29, 25, 4]-code or 29-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
(46, 54, large)-Net in Base 7 — Upper bound on s
There is no (46, 54, large)-net in base 7, because
- 6 times m-reduction [i] would yield (46, 48, large)-net in base 7, but