Best Known (59, 59+12, s)-Nets in Base 7
(59, 59+12, 137258)-Net over F7 — Constructive and digital
Digital (59, 71, 137258)-net over F7, using
- net defined by OOA [i] based on linear OOA(771, 137258, F7, 12, 12) (dual of [(137258, 12), 1647025, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(771, 823548, F7, 12) (dual of [823548, 823477, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(771, 823550, F7, 12) (dual of [823550, 823479, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [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(764, 823543, F7, 11) (dual of [823543, 823479, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 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(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(771, 823550, F7, 12) (dual of [823550, 823479, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(771, 823548, F7, 12) (dual of [823548, 823477, 13]-code), using
(59, 59+12, 621595)-Net over F7 — Digital
Digital (59, 71, 621595)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(771, 621595, F7, 12) (dual of [621595, 621524, 13]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(771, 823543, F7, 12) (dual of [823543, 823472, 13]-code), using
(59, 59+12, large)-Net in Base 7 — Upper bound on s
There is no (59, 71, large)-net in base 7, because
- 10 times m-reduction [i] would yield (59, 61, large)-net in base 7, but