Best Known (42, 42+17, s)-Nets in Base 5
(42, 42+17, 252)-Net over F5 — Constructive and digital
Digital (42, 59, 252)-net over F5, using
- 5 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+17, 689)-Net over F5 — Digital
Digital (42, 59, 689)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(559, 689, F5, 17) (dual of [689, 630, 18]-code), using
- 51 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 5 times 0, 1, 13 times 0, 1, 27 times 0) [i] based on linear OA(554, 633, F5, 17) (dual of [633, 579, 18]-code), using
- construction XX applied to C1 = C([141,156]), C2 = C([143,157]), C3 = C1 + C2 = C([143,156]), and C∩ = C1 ∩ C2 = C([141,157]) [i] based on
- linear OA(549, 624, F5, 16) (dual of [624, 575, 17]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {141,142,…,156}, and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(549, 624, F5, 15) (dual of [624, 575, 16]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {143,144,…,157}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(553, 624, F5, 17) (dual of [624, 571, 18]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {141,142,…,157}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(545, 624, F5, 14) (dual of [624, 579, 15]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {143,144,…,156}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- Reed–Solomon code RS(4,5) [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 XX applied to C1 = C([141,156]), C2 = C([143,157]), C3 = C1 + C2 = C([143,156]), and C∩ = C1 ∩ C2 = C([141,157]) [i] based on
- 51 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 5 times 0, 1, 13 times 0, 1, 27 times 0) [i] based on linear OA(554, 633, F5, 17) (dual of [633, 579, 18]-code), using
(42, 42+17, 109936)-Net in Base 5 — Upper bound on s
There is no (42, 59, 109937)-net in base 5, because
- 1 times m-reduction [i] would yield (42, 58, 109937)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 34696 388321 297771 668593 765287 623537 168865 > 558 [i]