Best Known (104, 138, s)-Nets in Base 5
(104, 138, 416)-Net over F5 — Constructive and digital
Digital (104, 138, 416)-net over F5, using
- trace code for nets [i] based on digital (35, 69, 208)-net over F25, using
- net from sequence [i] based on digital (35, 207)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 35 and N(F) ≥ 208, using
- net from sequence [i] based on digital (35, 207)-sequence over F25, using
(104, 138, 3120)-Net over F5 — Digital
Digital (104, 138, 3120)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5138, 3120, F5, 34) (dual of [3120, 2982, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(5138, 3138, F5, 34) (dual of [3138, 3000, 35]-code), using
- construction XX applied to Ce(33) ⊂ Ce(31) ⊂ Ce(30) [i] based on
- linear OA(5136, 3125, F5, 34) (dual of [3125, 2989, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(5126, 3125, F5, 32) (dual of [3125, 2999, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(5121, 3125, F5, 31) (dual of [3125, 3004, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(33) ⊂ Ce(31) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(5138, 3138, F5, 34) (dual of [3138, 3000, 35]-code), using
(104, 138, 846975)-Net in Base 5 — Upper bound on s
There is no (104, 138, 846976)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 2 869900 838063 870475 857065 250230 099175 901386 591542 138311 717423 438092 995758 797670 416816 410606 989825 > 5138 [i]