Best Known (45, 45+19, s)-Nets in Base 5
(45, 45+19, 252)-Net over F5 — Constructive and digital
Digital (45, 64, 252)-net over F5, using
- 6 times m-reduction [i] based on 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
(45, 45+19, 649)-Net over F5 — Digital
Digital (45, 64, 649)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(564, 649, F5, 19) (dual of [649, 585, 20]-code), using
- 14 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 10 times 0) [i] based on linear OA(561, 632, F5, 19) (dual of [632, 571, 20]-code), using
- construction XX applied to C1 = C([623,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([623,17]) [i] based on
- linear OA(557, 624, F5, 18) (dual of [624, 567, 19]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(557, 624, F5, 18) (dual of [624, 567, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(561, 624, F5, 19) (dual of [624, 563, 20]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,17}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(553, 624, F5, 17) (dual of [624, 571, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [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([623,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([623,17]) [i] based on
- 14 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 10 times 0) [i] based on linear OA(561, 632, F5, 19) (dual of [632, 571, 20]-code), using
(45, 45+19, 80993)-Net in Base 5 — Upper bound on s
There is no (45, 64, 80994)-net in base 5, because
- 1 times m-reduction [i] would yield (45, 63, 80994)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 108 431162 526099 396280 111982 836052 298799 958665 > 563 [i]