Best Known (45, 45+37, s)-Nets in Base 32
(45, 45+37, 240)-Net over F32 — Constructive and digital
Digital (45, 82, 240)-net over F32, using
- 9 times m-reduction [i] based on digital (45, 91, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 34, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 11 and N(F) ≥ 120, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- digital (11, 57, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 34, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(45, 45+37, 513)-Net in Base 32 — Constructive
(45, 82, 513)-net in base 32, using
- 20 times m-reduction [i] based on (45, 102, 513)-net in base 32, using
- base change [i] based on digital (28, 85, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- base change [i] based on digital (28, 85, 513)-net over F64, using
(45, 45+37, 1261)-Net over F32 — Digital
Digital (45, 82, 1261)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3282, 1261, F32, 37) (dual of [1261, 1179, 38]-code), using
- 222 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 35 times 0, 1, 60 times 0, 1, 90 times 0) [i] based on linear OA(3270, 1027, F32, 37) (dual of [1027, 957, 38]-code), using
- construction XX applied to C1 = C([1022,34]), C2 = C([0,35]), C3 = C1 + C2 = C([0,34]), and C∩ = C1 ∩ C2 = C([1022,35]) [i] based on
- linear OA(3268, 1023, F32, 36) (dual of [1023, 955, 37]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,34}, and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3268, 1023, F32, 36) (dual of [1023, 955, 37]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,35], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3270, 1023, F32, 37) (dual of [1023, 953, 38]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,35}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3266, 1023, F32, 35) (dual of [1023, 957, 36]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,34], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([1022,34]), C2 = C([0,35]), C3 = C1 + C2 = C([0,34]), and C∩ = C1 ∩ C2 = C([1022,35]) [i] based on
- 222 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 35 times 0, 1, 60 times 0, 1, 90 times 0) [i] based on linear OA(3270, 1027, F32, 37) (dual of [1027, 957, 38]-code), using
(45, 45+37, 1445241)-Net in Base 32 — Upper bound on s
There is no (45, 82, 1445242)-net in base 32, because
- 1 times m-reduction [i] would yield (45, 81, 1445242)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 82 632124 488929 986462 584475 476340 686645 957814 713760 423685 499329 824765 444409 227215 172210 226468 064973 389071 502877 687088 954660 > 3281 [i]