Best Known (14, 14+13, s)-Nets in Base 32
(14, 14+13, 172)-Net over F32 — Constructive and digital
Digital (14, 27, 172)-net over F32, using
- net defined by OOA [i] based on linear OOA(3227, 172, F32, 13, 13) (dual of [(172, 13), 2209, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3227, 1033, F32, 13) (dual of [1033, 1006, 14]-code), using
- construction XX applied to C1 = C([1020,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([1020,9]) [i] based on
- linear OA(3223, 1023, F32, 12) (dual of [1023, 1000, 13]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−3,−2,…,8}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(3219, 1023, F32, 10) (dual of [1023, 1004, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(3225, 1023, F32, 13) (dual of [1023, 998, 14]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−3,−2,…,9}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(3217, 1023, F32, 9) (dual of [1023, 1006, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(322, 8, F32, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), 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([1020,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([1020,9]) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(3227, 1033, F32, 13) (dual of [1033, 1006, 14]-code), using
(14, 14+13, 290)-Net in Base 32 — Constructive
(14, 27, 290)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 0 and N(F) ≥ 33, using
- the rational function field F32(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- (8, 21, 257)-net in base 32, using
- base change [i] based on (2, 15, 257)-net in base 128, using
- 1 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
- base change [i] based on digital (0, 14, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 14, 257)-net over F256, using
- 1 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
- base change [i] based on (2, 15, 257)-net in base 128, using
- digital (0, 6, 33)-net over F32, using
(14, 14+13, 567)-Net over F32 — Digital
Digital (14, 27, 567)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3227, 567, F32, 13) (dual of [567, 540, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3227, 1032, F32, 13) (dual of [1032, 1005, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(3225, 1024, F32, 13) (dual of [1024, 999, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(3219, 1024, F32, 10) (dual of [1024, 1005, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(322, 8, F32, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(3227, 1032, F32, 13) (dual of [1032, 1005, 14]-code), using
(14, 14+13, 321494)-Net in Base 32 — Upper bound on s
There is no (14, 27, 321495)-net in base 32, because
- 1 times m-reduction [i] would yield (14, 26, 321495)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 1361 146391 529185 460187 187327 420090 859365 > 3226 [i]