Best Known (59, 59+13, s)-Nets in Base 8
(59, 59+13, 43695)-Net over F8 — Constructive and digital
Digital (59, 72, 43695)-net over F8, using
- 81 times duplication [i] based on digital (58, 71, 43695)-net over F8, using
- net defined by OOA [i] based on linear OOA(871, 43695, F8, 13, 13) (dual of [(43695, 13), 567964, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(871, 262171, F8, 13) (dual of [262171, 262100, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(871, 262172, F8, 13) (dual of [262172, 262101, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(867, 262144, F8, 13) (dual of [262144, 262077, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(843, 262144, F8, 9) (dual of [262144, 262101, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(84, 28, F8, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,8)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(871, 262172, F8, 13) (dual of [262172, 262101, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(871, 262171, F8, 13) (dual of [262171, 262100, 14]-code), using
- net defined by OOA [i] based on linear OOA(871, 43695, F8, 13, 13) (dual of [(43695, 13), 567964, 14]-NRT-code), using
(59, 59+13, 262174)-Net over F8 — Digital
Digital (59, 72, 262174)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(872, 262174, F8, 13) (dual of [262174, 262102, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(871, 262172, F8, 13) (dual of [262172, 262101, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(867, 262144, F8, 13) (dual of [262144, 262077, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(843, 262144, F8, 9) (dual of [262144, 262101, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(84, 28, F8, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,8)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(871, 262173, F8, 12) (dual of [262173, 262102, 13]-code), using Gilbert–Varšamov bound and bm = 871 > Vbs−1(k−1) = 19 919736 703235 728401 330460 600398 153201 426623 138547 709174 107481 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(871, 262172, F8, 13) (dual of [262172, 262101, 14]-code), using
- construction X with Varšamov bound [i] based on
(59, 59+13, large)-Net in Base 8 — Upper bound on s
There is no (59, 72, large)-net in base 8, because
- 11 times m-reduction [i] would yield (59, 61, large)-net in base 8, but