Best Known (57, 57+23, s)-Nets in Base 5
(57, 57+23, 268)-Net over F5 — Constructive and digital
Digital (57, 80, 268)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (3, 14, 16)-net over F5, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 3 and N(F) ≥ 16, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- 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 (3, 14, 16)-net over F5, using
(57, 57+23, 800)-Net over F5 — Digital
Digital (57, 80, 800)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(580, 800, F5, 23) (dual of [800, 720, 24]-code), using
- 161 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0, 1, 19 times 0, 1, 31 times 0, 1, 43 times 0, 1, 51 times 0) [i] based on linear OA(573, 632, F5, 23) (dual of [632, 559, 24]-code), using
- construction XX applied to C1 = C([623,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([623,21]) [i] based on
- linear OA(569, 624, F5, 22) (dual of [624, 555, 23]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,20}, and designed minimum distance d ≥ |I|+1 = 23 [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(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 = {−1,0,…,21}, and designed minimum distance d ≥ |I|+1 = 24 [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(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([623,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([623,21]) [i] based on
- 161 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0, 1, 19 times 0, 1, 31 times 0, 1, 43 times 0, 1, 51 times 0) [i] based on linear OA(573, 632, F5, 23) (dual of [632, 559, 24]-code), using
(57, 57+23, 128470)-Net in Base 5 — Upper bound on s
There is no (57, 80, 128471)-net in base 5, because
- 1 times m-reduction [i] would yield (57, 79, 128471)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 16 544304 272984 389915 450295 329588 027687 484683 402202 878645 > 579 [i]