Best Known (103, 103+31, s)-Nets in Base 5
(103, 103+31, 504)-Net over F5 — Constructive and digital
Digital (103, 134, 504)-net over F5, using
- 52 times duplication [i] based on digital (101, 132, 504)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (35, 50, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 25, 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, 25, 126)-net over F25, using
- digital (51, 82, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 41, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25 (see above)
- trace code for nets [i] based on digital (10, 41, 126)-net over F25, using
- digital (35, 50, 252)-net over F5, using
- (u, u+v)-construction [i] based on
(103, 103+31, 4006)-Net over F5 — Digital
Digital (103, 134, 4006)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5134, 4006, F5, 31) (dual of [4006, 3872, 32]-code), using
- 867 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 17 times 0, 1, 32 times 0, 1, 54 times 0, 1, 85 times 0, 1, 120 times 0, 1, 155 times 0, 1, 181 times 0, 1, 201 times 0) [i] based on linear OA(5121, 3126, F5, 31) (dual of [3126, 3005, 32]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- 867 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 17 times 0, 1, 32 times 0, 1, 54 times 0, 1, 85 times 0, 1, 120 times 0, 1, 155 times 0, 1, 181 times 0, 1, 201 times 0) [i] based on linear OA(5121, 3126, F5, 31) (dual of [3126, 3005, 32]-code), using
(103, 103+31, 2530689)-Net in Base 5 — Upper bound on s
There is no (103, 134, 2530690)-net in base 5, because
- 1 times m-reduction [i] would yield (103, 133, 2530690)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 918 358321 557563 158703 858557 294914 351236 149966 650229 018914 098119 295666 908165 873360 723051 160105 > 5133 [i]