Best Known (57, 57+25, s)-Nets in Base 5
(57, 57+25, 258)-Net over F5 — Constructive and digital
Digital (57, 82, 258)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 12, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (45, 70, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 35, 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, 35, 126)-net over F25, using
- digital (0, 12, 6)-net over F5, using
(57, 57+25, 640)-Net over F5 — Digital
Digital (57, 82, 640)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(582, 640, F5, 25) (dual of [640, 558, 26]-code), using
- 7 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0) [i] based on linear OA(581, 632, F5, 25) (dual of [632, 551, 26]-code), using
- construction XX applied to C1 = C([623,22]), C2 = C([0,23]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([623,23]) [i] based on
- linear OA(577, 624, F5, 24) (dual of [624, 547, 25]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,22}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(577, 624, F5, 24) (dual of [624, 547, 25]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,23], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(581, 624, F5, 25) (dual of [624, 543, 26]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,23}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(573, 624, F5, 23) (dual of [624, 551, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [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
- linear OA(50, 4, F5, 0) (dual of [4, 4, 1]-code) (see above)
- construction XX applied to C1 = C([623,22]), C2 = C([0,23]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([623,23]) [i] based on
- 7 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0) [i] based on linear OA(581, 632, F5, 25) (dual of [632, 551, 26]-code), using
(57, 57+25, 69070)-Net in Base 5 — Upper bound on s
There is no (57, 82, 69071)-net in base 5, because
- 1 times m-reduction [i] would yield (57, 81, 69071)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 413 592951 097177 909867 588562 123286 018257 062971 446272 097745 > 581 [i]