Best Known (138−33, 138, s)-Nets in Base 5
(138−33, 138, 504)-Net over F5 — Constructive and digital
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
(138−33, 138, 3363)-Net over F5 — Digital
Digital (105, 138, 3363)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5138, 3363, F5, 33) (dual of [3363, 3225, 34]-code), using
- 226 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) [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
- 226 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) [i] based on linear OA(5131, 3130, F5, 33) (dual of [3130, 2999, 34]-code), using
(138−33, 138, 1642125)-Net in Base 5 — Upper bound on s
There is no (105, 138, 1642126)-net in base 5, because
- 1 times m-reduction [i] would yield (105, 137, 1642126)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 573974 459909 880662 373950 402960 594943 618111 574876 936742 319150 240339 488260 998938 504677 389471 098625 > 5137 [i]