Best Known (32−16, 32, s)-Nets in Base 32
(32−16, 32, 142)-Net over F32 — Constructive and digital
Digital (16, 32, 142)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- digital (7, 23, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (1, 9, 44)-net over F32, using
(32−16, 32, 261)-Net in Base 32 — Constructive
(16, 32, 261)-net in base 32, using
- base change [i] based on digital (4, 20, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
(32−16, 32, 515)-Net over F32 — Digital
Digital (16, 32, 515)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3232, 515, F32, 2, 16) (dual of [(515, 2), 998, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3232, 1030, F32, 16) (dual of [1030, 998, 17]-code), using
- construction XX applied to C1 = C([1021,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([1021,13]) [i] based on
- linear OA(3229, 1023, F32, 15) (dual of [1023, 994, 16]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,12}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(3227, 1023, F32, 14) (dual of [1023, 996, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(3231, 1023, F32, 16) (dual of [1023, 992, 17]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,13}, and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(3225, 1023, F32, 13) (dual of [1023, 998, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [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, 32, F32, 1) (dual of [32, 31, 2]-code), using
- Reed–Solomon code RS(31,32) [i]
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), 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([1021,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([1021,13]) [i] based on
- OOA 2-folding [i] based on linear OA(3232, 1030, F32, 16) (dual of [1030, 998, 17]-code), using
(32−16, 32, 127325)-Net in Base 32 — Upper bound on s
There is no (16, 32, 127326)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 1 461545 491487 962149 784803 996107 491855 126319 287321 > 3232 [i]