Best Known (39, 39+9, s)-Nets in Base 8
(39, 39+9, 65549)-Net over F8 — Constructive and digital
Digital (39, 48, 65549)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 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 (34, 43, 65535)-net over F8, using
- net defined by OOA [i] based on linear OOA(843, 65535, F8, 9, 9) (dual of [(65535, 9), 589772, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(843, 262141, F8, 9) (dual of [262141, 262098, 10]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(843, 262144, F8, 9) (dual of [262144, 262101, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(843, 262141, F8, 9) (dual of [262141, 262098, 10]-code), using
- net defined by OOA [i] based on linear OOA(843, 65535, F8, 9, 9) (dual of [(65535, 9), 589772, 10]-NRT-code), using
- digital (1, 5, 14)-net over F8, using
(39, 39+9, 262168)-Net over F8 — Digital
Digital (39, 48, 262168)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(848, 262168, F8, 9) (dual of [262168, 262120, 10]-code), using
- construction XX applied to Ce(8) ⊂ Ce(4) ⊂ Ce(3) [i] based on
- 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(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(819, 262144, F8, 4) (dual of [262144, 262125, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(8) ⊂ Ce(4) ⊂ Ce(3) [i] based on
(39, 39+9, large)-Net in Base 8 — Upper bound on s
There is no (39, 48, large)-net in base 8, because
- 7 times m-reduction [i] would yield (39, 41, large)-net in base 8, but