Best Known (137−34, 137, s)-Nets in Base 5
(137−34, 137, 412)-Net over F5 — Constructive and digital
Digital (103, 137, 412)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 19, 12)-net over F5, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 2 and N(F) ≥ 12, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- digital (84, 118, 400)-net over F5, using
- trace code for nets [i] based on digital (25, 59, 200)-net over F25, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 25 and N(F) ≥ 200, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- trace code for nets [i] based on digital (25, 59, 200)-net over F25, using
- digital (2, 19, 12)-net over F5, using
(137−34, 137, 2966)-Net over F5 — Digital
Digital (103, 137, 2966)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5137, 2966, F5, 34) (dual of [2966, 2829, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(5137, 3136, F5, 34) (dual of [3136, 2999, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(31) [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(51, 11, F5, 1) (dual of [11, 10, 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 X applied to Ce(33) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(5137, 3136, F5, 34) (dual of [3136, 2999, 35]-code), using
(137−34, 137, 770467)-Net in Base 5 — Upper bound on s
There is no (103, 137, 770468)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 573980 343480 689032 950403 015919 376171 025141 487073 793180 449846 305006 529365 799375 232465 000181 151889 > 5137 [i]