Best Known (45, 45+10, s)-Nets in Base 8
(45, 45+10, 52444)-Net over F8 — Constructive and digital
Digital (45, 55, 52444)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 14)-net over F8, using
- net from sequence [i] based on digital (1, 13)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 1 and N(F) ≥ 14, using
- net from sequence [i] based on digital (1, 13)-sequence over F8, using
- digital (39, 49, 52430)-net over F8, using
- net defined by OOA [i] based on linear OOA(849, 52430, F8, 10, 10) (dual of [(52430, 10), 524251, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(849, 262150, F8, 10) (dual of [262150, 262101, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(849, 262144, F8, 10) (dual of [262144, 262095, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(80, 6, F8, 0) (dual of [6, 6, 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
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- OA 5-folding and stacking [i] based on linear OA(849, 262150, F8, 10) (dual of [262150, 262101, 11]-code), using
- net defined by OOA [i] based on linear OOA(849, 52430, F8, 10, 10) (dual of [(52430, 10), 524251, 11]-NRT-code), using
- digital (1, 6, 14)-net over F8, using
(45, 45+10, 262174)-Net over F8 — Digital
Digital (45, 55, 262174)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(855, 262174, F8, 10) (dual of [262174, 262119, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
- linear OA(849, 262144, F8, 10) (dual of [262144, 262095, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(825, 262144, F8, 5) (dual of [262144, 262119, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(86, 30, F8, 4) (dual of [30, 24, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- 1 times truncation [i] based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
(45, 45+10, large)-Net in Base 8 — Upper bound on s
There is no (45, 55, large)-net in base 8, because
- 8 times m-reduction [i] would yield (45, 47, large)-net in base 8, but