Best Known (54, 54+23, s)-Nets in Base 5
(54, 54+23, 258)-Net over F5 — Constructive and digital
Digital (54, 77, 258)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 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 (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 (0, 11, 6)-net over F5, using
(54, 54+23, 669)-Net over F5 — Digital
Digital (54, 77, 669)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(577, 669, F5, 23) (dual of [669, 592, 24]-code), using
- 33 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0, 1, 19 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
- 33 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0, 1, 19 times 0) [i] based on linear OA(573, 632, F5, 23) (dual of [632, 559, 24]-code), using
(54, 54+23, 82824)-Net in Base 5 — Upper bound on s
There is no (54, 77, 82825)-net in base 5, because
- 1 times m-reduction [i] would yield (54, 76, 82825)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 132350 946147 355957 487250 738907 180029 490560 885650 146941 > 576 [i]