Best Known (81−24, 81, s)-Nets in Base 5
(81−24, 81, 262)-Net over F5 — Constructive and digital
Digital (57, 81, 262)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- digital (44, 68, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 34, 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, 34, 126)-net over F25, using
- digital (1, 13, 10)-net over F5, using
(81−24, 81, 698)-Net over F5 — Digital
Digital (57, 81, 698)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(581, 698, F5, 24) (dual of [698, 617, 25]-code), using
- 58 step Varšamov–Edel lengthening with (ri) = (1, 4 times 0, 1, 17 times 0, 1, 34 times 0) [i] based on linear OA(578, 637, F5, 24) (dual of [637, 559, 25]-code), using
- construction XX applied to C1 = C([622,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([622,21]) [i] based on
- linear OA(573, 624, F5, 23) (dual of [624, 551, 24]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−2,−1,…,20}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(569, 624, F5, 22) (dual of [624, 555, 23]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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 = {−2,−1,…,21}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(565, 624, F5, 21) (dual of [624, 559, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- 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([622,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([622,21]) [i] based on
- 58 step Varšamov–Edel lengthening with (ri) = (1, 4 times 0, 1, 17 times 0, 1, 34 times 0) [i] based on linear OA(578, 637, F5, 24) (dual of [637, 559, 25]-code), using
(81−24, 81, 69070)-Net in Base 5 — Upper bound on s
There is no (57, 81, 69071)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 413 592951 097177 909867 588562 123286 018257 062971 446272 097745 > 581 [i]