Best Known (90−45, 90, s)-Nets in Base 32
(90−45, 90, 240)-Net over F32 — Constructive and digital
Digital (45, 90, 240)-net over F32, using
- 1 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
(90−45, 90, 513)-Net in Base 32 — Constructive
(45, 90, 513)-net in base 32, using
- 12 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
(90−45, 90, 691)-Net over F32 — Digital
Digital (45, 90, 691)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3290, 691, F32, 45) (dual of [691, 601, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3290, 1030, F32, 45) (dual of [1030, 940, 46]-code), using
- construction X applied to C([0,22]) ⊂ C([0,21]) [i] based on
- linear OA(3289, 1025, F32, 45) (dual of [1025, 936, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,22], and minimum distance d ≥ |{−22,−21,…,22}|+1 = 46 (BCH-bound) [i]
- linear OA(3285, 1025, F32, 43) (dual of [1025, 940, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(321, 5, F32, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,22]) ⊂ C([0,21]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3290, 1030, F32, 45) (dual of [1030, 940, 46]-code), using
(90−45, 90, 358503)-Net in Base 32 — Upper bound on s
There is no (45, 90, 358504)-net in base 32, because
- 1 times m-reduction [i] would yield (45, 89, 358504)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 90 856821 276530 606339 012661 407569 709047 366063 350318 932508 736186 842254 651256 796856 445515 359102 695393 899856 486714 282227 453834 192460 790214 > 3289 [i]