Best Known (137, 137+31, s)-Nets in Base 8
(137, 137+31, 17478)-Net over F8 — Constructive and digital
Digital (137, 168, 17478)-net over F8, using
- 81 times duplication [i] based on digital (136, 167, 17478)-net over F8, using
- net defined by OOA [i] based on linear OOA(8167, 17478, F8, 31, 31) (dual of [(17478, 31), 541651, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(8167, 262171, F8, 31) (dual of [262171, 262004, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(8167, 262172, F8, 31) (dual of [262172, 262005, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(8163, 262144, F8, 31) (dual of [262144, 261981, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(8139, 262144, F8, 27) (dual of [262144, 262005, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(30) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(8167, 262172, F8, 31) (dual of [262172, 262005, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(8167, 262171, F8, 31) (dual of [262171, 262004, 32]-code), using
- net defined by OOA [i] based on linear OOA(8167, 17478, F8, 31, 31) (dual of [(17478, 31), 541651, 32]-NRT-code), using
(137, 137+31, 262174)-Net over F8 — Digital
Digital (137, 168, 262174)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8168, 262174, F8, 31) (dual of [262174, 262006, 32]-code), using
- construction XX applied to Ce(30) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- linear OA(8163, 262144, F8, 31) (dual of [262144, 261981, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(8139, 262144, F8, 27) (dual of [262144, 262005, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(8133, 262144, F8, 26) (dual of [262144, 262011, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(84, 29, F8, 3) (dual of [29, 25, 4]-code or 29-cap in PG(3,8)), using
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(30) ⊂ Ce(26) ⊂ Ce(25) [i] based on
(137, 137+31, large)-Net in Base 8 — Upper bound on s
There is no (137, 168, large)-net in base 8, because
- 29 times m-reduction [i] would yield (137, 139, large)-net in base 8, but