Best Known (55−15, 55, s)-Nets in Base 5
(55−15, 55, 252)-Net over F5 — Constructive and digital
Digital (40, 55, 252)-net over F5, using
- 5 times m-reduction [i] based on digital (40, 60, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 30, 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, 30, 126)-net over F25, using
(55−15, 55, 853)-Net over F5 — Digital
Digital (40, 55, 853)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(555, 853, F5, 15) (dual of [853, 798, 16]-code), using
- 222 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0, 1, 22 times 0, 1, 40 times 0, 1, 62 times 0, 1, 81 times 0) [i] based on linear OA(548, 624, F5, 15) (dual of [624, 576, 16]-code), using
- 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]
- 222 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0, 1, 22 times 0, 1, 40 times 0, 1, 62 times 0, 1, 81 times 0) [i] based on linear OA(548, 624, F5, 15) (dual of [624, 576, 16]-code), using
(55−15, 55, 208402)-Net in Base 5 — Upper bound on s
There is no (40, 55, 208403)-net in base 5, because
- 1 times m-reduction [i] would yield (40, 54, 208403)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 55 511946 591866 289729 034324 298595 923445 > 554 [i]