Best Known (15, 27, s)-Nets in Base 32
(15, 27, 173)-Net over F32 — Constructive and digital
Digital (15, 27, 173)-net over F32, using
- net defined by OOA [i] based on linear OOA(3227, 173, F32, 12, 12) (dual of [(173, 12), 2049, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(3227, 1038, F32, 12) (dual of [1038, 1011, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- linear OA(3223, 1024, F32, 12) (dual of [1024, 1001, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(3213, 1024, F32, 7) (dual of [1024, 1011, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(324, 14, F32, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,32)), using
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- Reed–Solomon code RS(28,32) [i]
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- OA 6-folding and stacking [i] based on linear OA(3227, 1038, F32, 12) (dual of [1038, 1011, 13]-code), using
(15, 27, 301)-Net in Base 32 — Constructive
(15, 27, 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, 20, 257)-net in base 32, using
- 1 times m-reduction [i] based on (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
- 1 times m-reduction [i] based on (8, 21, 257)-net in base 32, using
- digital (1, 7, 44)-net over F32, using
(15, 27, 1069)-Net over F32 — Digital
Digital (15, 27, 1069)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3227, 1069, F32, 12) (dual of [1069, 1042, 13]-code), using
- 38 step Varšamov–Edel lengthening with (ri) = (3, 5 times 0, 1, 31 times 0) [i] based on linear OA(3223, 1027, F32, 12) (dual of [1027, 1004, 13]-code), using
- construction XX applied to C1 = C([1022,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([1022,10]) [i] based on
- linear OA(3221, 1023, F32, 11) (dual of [1023, 1002, 12]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(3221, 1023, F32, 11) (dual of [1023, 1002, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- 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 = {−1,0,…,10}, 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(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
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([1022,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([1022,10]) [i] based on
- 38 step Varšamov–Edel lengthening with (ri) = (3, 5 times 0, 1, 31 times 0) [i] based on linear OA(3223, 1027, F32, 12) (dual of [1027, 1004, 13]-code), using
(15, 27, 572839)-Net in Base 32 — Upper bound on s
There is no (15, 27, 572840)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 43556 195204 312760 282259 420652 723936 615359 > 3227 [i]