Best Known (139−34, 139, s)-Nets in Base 5
(139−34, 139, 418)-Net over F5 — Constructive and digital
Digital (105, 139, 418)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (4, 21, 18)-net over F5, using
- net from sequence [i] based on digital (4, 17)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 4 and N(F) ≥ 18, using
- net from sequence [i] based on digital (4, 17)-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 (4, 21, 18)-net over F5, using
(139−34, 139, 3152)-Net over F5 — Digital
Digital (105, 139, 3152)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5139, 3152, F5, 34) (dual of [3152, 3013, 35]-code), using
- 19 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 14 times 0) [i] based on linear OA(5136, 3130, F5, 34) (dual of [3130, 2994, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(32) [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(5131, 3125, F5, 33) (dual of [3125, 2994, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(33) ⊂ Ce(32) [i] based on
- 19 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 14 times 0) [i] based on linear OA(5136, 3130, F5, 34) (dual of [3130, 2994, 35]-code), using
(139−34, 139, 931080)-Net in Base 5 — Upper bound on s
There is no (105, 139, 931081)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 14 349453 681890 869611 986869 084733 223930 540970 889804 049130 997071 548876 098225 148747 846821 682066 500325 > 5139 [i]