Best Known (55, 106, s)-Nets in Base 32
(55, 106, 240)-Net over F32 — Constructive and digital
Digital (55, 106, 240)-net over F32, using
- t-expansion [i] based on digital (51, 106, 240)-net over F32, using
- 3 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 40, 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, 69, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 40, 120)-net over F32, using
- (u, u+v)-construction [i] based on
- 3 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
(55, 106, 513)-Net in Base 32 — Constructive
(55, 106, 513)-net in base 32, using
- t-expansion [i] based on (46, 106, 513)-net in base 32, using
- 2 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
- base change [i] based on digital (28, 90, 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, 90, 513)-net over F64, using
- 2 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(55, 106, 1013)-Net over F32 — Digital
Digital (55, 106, 1013)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(32106, 1013, F32, 51) (dual of [1013, 907, 52]-code), using
- discarding factors / shortening the dual code based on linear OA(32106, 1042, F32, 51) (dual of [1042, 936, 52]-code), using
- construction X applied to C([0,25]) ⊂ C([0,22]) [i] based on
- linear OA(32101, 1025, F32, 51) (dual of [1025, 924, 52]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,25], and minimum distance d ≥ |{−25,−24,…,25}|+1 = 52 (BCH-bound) [i]
- 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(325, 17, F32, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,32)), using
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- Reed–Solomon code RS(27,32) [i]
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- construction X applied to C([0,25]) ⊂ C([0,22]) [i] based on
- discarding factors / shortening the dual code based on linear OA(32106, 1042, F32, 51) (dual of [1042, 936, 52]-code), using
(55, 106, 688471)-Net in Base 32 — Upper bound on s
There is no (55, 106, 688472)-net in base 32, because
- 1 times m-reduction [i] would yield (55, 105, 688472)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 109 838775 421849 973335 513825 533702 158657 920880 797168 090786 939205 117344 097472 001831 698035 621467 098779 187396 762966 777479 113670 603175 110865 688558 201100 801695 574040 > 32105 [i]