Best Known (44, 44+19, s)-Nets in Base 5
(44, 44+19, 252)-Net over F5 — Constructive and digital
Digital (44, 63, 252)-net over F5, using
- 5 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+19, 624)-Net over F5 — Digital
Digital (44, 63, 624)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(563, 624, F5, 19) (dual of [624, 561, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(563, 636, F5, 19) (dual of [636, 573, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- linear OA(561, 625, F5, 19) (dual of [625, 564, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(51, 10, F5, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(563, 636, F5, 19) (dual of [636, 573, 20]-code), using
(44, 44+19, 67729)-Net in Base 5 — Upper bound on s
There is no (44, 63, 67730)-net in base 5, because
- 1 times m-reduction [i] would yield (44, 62, 67730)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 21 685389 322742 625205 145373 097509 638152 869705 > 562 [i]