Best Known (40−9, 40, s)-Nets in Base 5
(40−9, 40, 788)-Net over F5 — Constructive and digital
Digital (31, 40, 788)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (27, 36, 782)-net over F5, using
- net defined by OOA [i] based on linear OOA(536, 782, F5, 9, 9) (dual of [(782, 9), 7002, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(536, 3129, F5, 9) (dual of [3129, 3093, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(536, 3130, F5, 9) (dual of [3130, 3094, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(536, 3125, F5, 9) (dual of [3125, 3089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(531, 3125, F5, 8) (dual of [3125, 3094, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(536, 3130, F5, 9) (dual of [3130, 3094, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(536, 3129, F5, 9) (dual of [3129, 3093, 10]-code), using
- net defined by OOA [i] based on linear OOA(536, 782, F5, 9, 9) (dual of [(782, 9), 7002, 10]-NRT-code), using
- digital (0, 4, 6)-net over F5, using
(40−9, 40, 3327)-Net over F5 — Digital
Digital (31, 40, 3327)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(540, 3327, F5, 9) (dual of [3327, 3287, 10]-code), using
- 187 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0, 1, 34 times 0, 1, 144 times 0) [i] based on linear OA(537, 3137, F5, 9) (dual of [3137, 3100, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(536, 3125, F5, 9) (dual of [3125, 3089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(526, 3125, F5, 7) (dual of [3125, 3099, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(511, 12, F5, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,5)), using
- dual of repetition code with length 12 [i]
- linear OA(51, 12, F5, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- 187 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0, 1, 34 times 0, 1, 144 times 0) [i] based on linear OA(537, 3137, F5, 9) (dual of [3137, 3100, 10]-code), using
(40−9, 40, 3613683)-Net in Base 5 — Upper bound on s
There is no (31, 40, 3613684)-net in base 5, because
- 1 times m-reduction [i] would yield (31, 39, 3613684)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 1818 991278 445186 989639 674625 > 539 [i]