Best Known (103−23, 103, s)-Nets in Base 5
(103−23, 103, 408)-Net over F5 — Constructive and digital
Digital (80, 103, 408)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (26, 37, 156)-net over F5, using
- generalized (u, u+v)-construction [i] based on
- digital (2, 5, 52)-net over F5, using
- s-reduction based on digital (2, 5, 66)-net over F5, using
- net defined by OOA [i] based on linear OOA(55, 66, F5, 3, 3) (dual of [(66, 3), 193, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(55, 66, F5, 2, 3) (dual of [(66, 2), 127, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(55, 66, F5, 3, 3) (dual of [(66, 3), 193, 4]-NRT-code), using
- s-reduction based on digital (2, 5, 66)-net over F5, using
- digital (5, 10, 52)-net over F5, using
- s-reduction based on digital (5, 10, 68)-net over F5, using
- digital (11, 22, 52)-net over F5, using
- trace code for nets [i] based on digital (0, 11, 26)-net over F25, using
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using
- the rational function field F25(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- trace code for nets [i] based on digital (0, 11, 26)-net over F25, using
- digital (2, 5, 52)-net over F5, using
- generalized (u, u+v)-construction [i] based on
- digital (43, 66, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 33, 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, 33, 126)-net over F25, using
- digital (26, 37, 156)-net over F5, using
(103−23, 103, 4252)-Net over F5 — Digital
Digital (80, 103, 4252)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5103, 4252, F5, 23) (dual of [4252, 4149, 24]-code), using
- 1110 step Varšamov–Edel lengthening with (ri) = (2, 1, 1, 0, 0, 0, 1, 10 times 0, 1, 23 times 0, 1, 48 times 0, 1, 93 times 0, 1, 155 times 0, 1, 215 times 0, 1, 260 times 0, 1, 292 times 0) [i] based on linear OA(591, 3130, F5, 23) (dual of [3130, 3039, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(591, 3125, F5, 23) (dual of [3125, 3034, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(586, 3125, F5, 22) (dual of [3125, 3039, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 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 X applied to Ce(22) ⊂ Ce(21) [i] based on
- 1110 step Varšamov–Edel lengthening with (ri) = (2, 1, 1, 0, 0, 0, 1, 10 times 0, 1, 23 times 0, 1, 48 times 0, 1, 93 times 0, 1, 155 times 0, 1, 215 times 0, 1, 260 times 0, 1, 292 times 0) [i] based on linear OA(591, 3130, F5, 23) (dual of [3130, 3039, 24]-code), using
(103−23, 103, 3718017)-Net in Base 5 — Upper bound on s
There is no (80, 103, 3718018)-net in base 5, because
- 1 times m-reduction [i] would yield (80, 102, 3718018)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 197215 258099 495011 586221 553636 008382 888869 438066 852817 673924 013378 502569 > 5102 [i]