Best Known (47, 47+42, s)-Nets in Base 32
(47, 47+42, 240)-Net over F32 — Constructive and digital
Digital (47, 89, 240)-net over F32, using
- 8 times m-reduction [i] based on digital (47, 97, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 36, 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, 61, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 36, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(47, 47+42, 513)-Net in Base 32 — Constructive
(47, 89, 513)-net in base 32, using
- t-expansion [i] based on (46, 89, 513)-net in base 32, using
- 19 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
- 19 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(47, 47+42, 1023)-Net over F32 — Digital
Digital (47, 89, 1023)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3289, 1023, F32, 42) (dual of [1023, 934, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3289, 1051, F32, 42) (dual of [1051, 962, 43]-code), using
- construction XX applied to C1 = C([1016,33]), C2 = C([3,34]), C3 = C1 + C2 = C([3,33]), and C∩ = C1 ∩ C2 = C([1016,34]) [i] based on
- linear OA(3278, 1023, F32, 41) (dual of [1023, 945, 42]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−7,−6,…,33}, and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(3263, 1023, F32, 32) (dual of [1023, 960, 33]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {3,4,…,34}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3280, 1023, F32, 42) (dual of [1023, 943, 43]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−7,−6,…,34}, and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(3261, 1023, F32, 31) (dual of [1023, 962, 32]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {3,4,…,33}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(329, 26, F32, 9) (dual of [26, 17, 10]-code or 26-arc in PG(8,32)), using
- discarding factors / shortening the dual code based on linear OA(329, 32, F32, 9) (dual of [32, 23, 10]-code or 32-arc in PG(8,32)), using
- Reed–Solomon code RS(23,32) [i]
- discarding factors / shortening the dual code based on linear OA(329, 32, F32, 9) (dual of [32, 23, 10]-code or 32-arc in PG(8,32)), using
- 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
- construction XX applied to C1 = C([1016,33]), C2 = C([3,34]), C3 = C1 + C2 = C([3,33]), and C∩ = C1 ∩ C2 = C([1016,34]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3289, 1051, F32, 42) (dual of [1051, 962, 43]-code), using
(47, 47+42, 670027)-Net in Base 32 — Upper bound on s
There is no (47, 89, 670028)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 90 855031 913291 640173 234780 920312 012819 379656 869035 501807 142054 121184 472023 286515 822932 659833 339287 364103 304036 865455 897287 881441 196504 > 3289 [i]