Best Known (63, 75, s)-Nets in Base 7
(63, 75, 137262)-Net over F7 — Constructive and digital
Digital (63, 75, 137262)-net over F7, using
- net defined by OOA [i] based on linear OOA(775, 137262, F7, 12, 12) (dual of [(137262, 12), 1647069, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(775, 823572, F7, 12) (dual of [823572, 823497, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(775, 823575, F7, 12) (dual of [823575, 823500, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(771, 823543, F7, 12) (dual of [823543, 823472, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(74, 32, F7, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,7)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(775, 823575, F7, 12) (dual of [823575, 823500, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(775, 823572, F7, 12) (dual of [823572, 823497, 13]-code), using
(63, 75, 823575)-Net over F7 — Digital
Digital (63, 75, 823575)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(775, 823575, F7, 12) (dual of [823575, 823500, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(771, 823543, F7, 12) (dual of [823543, 823472, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(74, 32, F7, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,7)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
(63, 75, large)-Net in Base 7 — Upper bound on s
There is no (63, 75, large)-net in base 7, because
- 10 times m-reduction [i] would yield (63, 65, large)-net in base 7, but