Best Known (42, 42+18, s)-Nets in Base 5
(42, 42+18, 252)-Net over F5 — Constructive and digital
Digital (42, 60, 252)-net over F5, using
- 4 times m-reduction [i] based on digital (42, 64, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 32, 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, 32, 126)-net over F25, using
(42, 42+18, 632)-Net over F5 — Digital
Digital (42, 60, 632)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(560, 632, F5, 18) (dual of [632, 572, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(560, 637, F5, 18) (dual of [637, 577, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(557, 625, F5, 18) (dual of [625, 568, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(545, 625, F5, 14) (dual of [625, 580, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(51, 2, F5, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(17) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(560, 637, F5, 18) (dual of [637, 577, 19]-code), using
(42, 42+18, 47362)-Net in Base 5 — Upper bound on s
There is no (42, 60, 47363)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 867451 976356 433718 294299 911807 428869 389805 > 560 [i]