Best Known (24, 24+12, s)-Nets in Base 5
(24, 24+12, 132)-Net over F5 — Constructive and digital
Digital (24, 36, 132)-net over F5, using
- 4 times m-reduction [i] based on digital (24, 40, 132)-net over F5, using
- trace code for nets [i] based on digital (4, 20, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- trace code for nets [i] based on digital (4, 20, 66)-net over F25, using
(24, 24+12, 244)-Net over F5 — Digital
Digital (24, 36, 244)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(536, 244, F5, 12) (dual of [244, 208, 13]-code), using
- 108 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 14 times 0, 1, 20 times 0, 1, 25 times 0, 1, 30 times 0) [i] based on linear OA(528, 128, F5, 12) (dual of [128, 100, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(528, 125, F5, 12) (dual of [125, 97, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(525, 125, F5, 11) (dual of [125, 100, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(50, 3, F5, 0) (dual of [3, 3, 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(11) ⊂ Ce(10) [i] based on
- 108 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 14 times 0, 1, 20 times 0, 1, 25 times 0, 1, 30 times 0) [i] based on linear OA(528, 128, F5, 12) (dual of [128, 100, 13]-code), using
(24, 24+12, 11690)-Net in Base 5 — Upper bound on s
There is no (24, 36, 11691)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 14 555552 375527 615141 951465 > 536 [i]