Best Known (44, 44+17, s)-Nets in Base 5
(44, 44+17, 252)-Net over F5 — Constructive and digital
Digital (44, 61, 252)-net over F5, using
- 7 times m-reduction [i] based on digital (44, 68, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 34, 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, 34, 126)-net over F25, using
(44, 44+17, 800)-Net over F5 — Digital
Digital (44, 61, 800)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(561, 800, F5, 17) (dual of [800, 739, 18]-code), using
- 163 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 6 times 0, 1, 13 times 0, 1, 27 times 0, 1, 45 times 0, 1, 63 times 0) [i] based on linear OA(553, 629, F5, 17) (dual of [629, 576, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(553, 625, F5, 17) (dual of [625, 572, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(549, 625, F5, 16) (dual of [625, 576, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(50, 4, F5, 0) (dual of [4, 4, 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(16) ⊂ Ce(15) [i] based on
- 163 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 6 times 0, 1, 13 times 0, 1, 27 times 0, 1, 45 times 0, 1, 63 times 0) [i] based on linear OA(553, 629, F5, 17) (dual of [629, 576, 18]-code), using
(44, 44+17, 164395)-Net in Base 5 — Upper bound on s
There is no (44, 61, 164396)-net in base 5, because
- 1 times m-reduction [i] would yield (44, 60, 164396)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 867372 425687 729282 804708 246988 782198 431745 > 560 [i]