Best Known (54−10, 54, s)-Nets in Base 8
(54−10, 54, 52439)-Net over F8 — Constructive and digital
Digital (44, 54, 52439)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 9)-net over F8, using
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 0 and N(F) ≥ 9, using
- the rational function field F8(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 8)-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 (0, 5, 9)-net over F8, using
(54−10, 54, 262168)-Net over F8 — Digital
Digital (44, 54, 262168)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(854, 262168, F8, 10) (dual of [262168, 262114, 11]-code), using
- construction XX applied to Ce(9) ⊂ Ce(5) ⊂ 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(831, 262144, F8, 6) (dual of [262144, 262113, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(84, 23, F8, 3) (dual of [23, 19, 4]-code or 23-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(9) ⊂ Ce(5) ⊂ Ce(4) [i] based on
(54−10, 54, large)-Net in Base 8 — Upper bound on s
There is no (44, 54, large)-net in base 8, because
- 8 times m-reduction [i] would yield (44, 46, large)-net in base 8, but