Best Known (139−33, 139, s)-Nets in Base 5
(139−33, 139, 504)-Net over F5 — Constructive and digital
Digital (106, 139, 504)-net over F5, using
- 51 times duplication [i] based on digital (105, 138, 504)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (36, 52, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 26, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 26, 126)-net over F25, using
- digital (53, 86, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 43, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25 (see above)
- trace code for nets [i] based on digital (10, 43, 126)-net over F25, using
- digital (36, 52, 252)-net over F5, using
- (u, u+v)-construction [i] based on
(139−33, 139, 3506)-Net over F5 — Digital
Digital (106, 139, 3506)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5139, 3506, F5, 33) (dual of [3506, 3367, 34]-code), using
- 368 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 14 times 0, 1, 32 times 0, 1, 63 times 0, 1, 103 times 0, 1, 141 times 0) [i] based on linear OA(5131, 3130, F5, 33) (dual of [3130, 2999, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(31) [i] based on
- 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(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(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(32) ⊂ Ce(31) [i] based on
- 368 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 14 times 0, 1, 32 times 0, 1, 63 times 0, 1, 103 times 0, 1, 141 times 0) [i] based on linear OA(5131, 3130, F5, 33) (dual of [3130, 2999, 34]-code), using
(139−33, 139, 1815901)-Net in Base 5 — Upper bound on s
There is no (106, 139, 1815902)-net in base 5, because
- 1 times m-reduction [i] would yield (106, 138, 1815902)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 2 869873 699846 577114 292253 643767 789305 467475 531875 320860 059369 610222 305740 496913 454399 641285 641985 > 5138 [i]