Best Known (28−13, 28, s)-Nets in Base 32
(28−13, 28, 172)-Net over F32 — Constructive and digital
Digital (15, 28, 172)-net over F32, using
- 321 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(3227, 172, F32, 13, 13) (dual of [(172, 13), 2209, 14]-NRT-code), using
(28−13, 28, 301)-Net in Base 32 — Constructive
(15, 28, 301)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 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
- (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 (1, 7, 44)-net over F32, using
(28−13, 28, 779)-Net over F32 — Digital
Digital (15, 28, 779)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3228, 779, F32, 13) (dual of [779, 751, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3228, 1036, F32, 13) (dual of [1036, 1008, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(3225, 1025, F32, 13) (dual of [1025, 1000, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(3217, 1025, F32, 9) (dual of [1025, 1008, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(323, 11, F32, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,32) or 11-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3228, 1036, F32, 13) (dual of [1036, 1008, 14]-code), using
(28−13, 28, 572839)-Net in Base 32 — Upper bound on s
There is no (15, 28, 572840)-net in base 32, because
- 1 times m-reduction [i] would yield (15, 27, 572840)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 43556 195204 312760 282259 420652 723936 615359 > 3227 [i]