Best Known (32, 32+9, s)-Nets in Base 8
(32, 32+9, 8205)-Net over F8 — Constructive and digital
Digital (32, 41, 8205)-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 (27, 36, 8191)-net over F8, using
- net defined by OOA [i] based on linear OOA(836, 8191, F8, 9, 9) (dual of [(8191, 9), 73683, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(836, 32765, F8, 9) (dual of [32765, 32729, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(836, 32765, F8, 9) (dual of [32765, 32729, 10]-code), using
- net defined by OOA [i] based on linear OOA(836, 8191, F8, 9, 9) (dual of [(8191, 9), 73683, 10]-NRT-code), using
- digital (1, 5, 14)-net over F8, using
(32, 32+9, 32789)-Net over F8 — Digital
Digital (32, 41, 32789)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(841, 32789, F8, 9) (dual of [32789, 32748, 10]-code), using
- construction XX applied to Ce(8) ⊂ Ce(4) ⊂ Ce(3) [i] based on
- linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(821, 32768, F8, 5) (dual of [32768, 32747, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(816, 32768, F8, 4) (dual of [32768, 32752, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(84, 20, F8, 3) (dual of [20, 16, 4]-code or 20-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
(32, 32+9, large)-Net in Base 8 — Upper bound on s
There is no (32, 41, large)-net in base 8, because
- 7 times m-reduction [i] would yield (32, 34, large)-net in base 8, but