Best Known (27, 27+20, s)-Nets in Base 32
(27, 27+20, 206)-Net over F32 — Constructive and digital
Digital (27, 47, 206)-net over F32, using
- generalized (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
- digital (3, 13, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- digital (7, 27, 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, 7, 44)-net over F32, using
(27, 27+20, 409)-Net in Base 32 — Constructive
(27, 47, 409)-net in base 32, using
- net defined by OOA [i] based on OOA(3247, 409, S32, 20, 20), using
- OA 10-folding and stacking [i] based on OA(3247, 4090, S32, 20), using
- discarding factors based on OA(3247, 4098, S32, 20), using
- discarding parts of the base [i] based on linear OA(6439, 4098, F64, 20) (dual of [4098, 4059, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(6439, 4096, F64, 20) (dual of [4096, 4057, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(6437, 4096, F64, 19) (dual of [4096, 4059, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- discarding parts of the base [i] based on linear OA(6439, 4098, F64, 20) (dual of [4098, 4059, 21]-code), using
- discarding factors based on OA(3247, 4098, S32, 20), using
- OA 10-folding and stacking [i] based on OA(3247, 4090, S32, 20), using
(27, 27+20, 1370)-Net over F32 — Digital
Digital (27, 47, 1370)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3247, 1370, F32, 20) (dual of [1370, 1323, 21]-code), using
- 335 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 10 times 0, 1, 36 times 0, 1, 96 times 0, 1, 185 times 0) [i] based on linear OA(3239, 1027, F32, 20) (dual of [1027, 988, 21]-code), using
- construction XX applied to C1 = C([1022,17]), C2 = C([0,18]), C3 = C1 + C2 = C([0,17]), and C∩ = C1 ∩ C2 = C([1022,18]) [i] based on
- linear OA(3237, 1023, F32, 19) (dual of [1023, 986, 20]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,17}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3237, 1023, F32, 19) (dual of [1023, 986, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3239, 1023, F32, 20) (dual of [1023, 984, 21]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,18}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(3235, 1023, F32, 18) (dual of [1023, 988, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [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([1022,17]), C2 = C([0,18]), C3 = C1 + C2 = C([0,17]), and C∩ = C1 ∩ C2 = C([1022,18]) [i] based on
- 335 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 10 times 0, 1, 36 times 0, 1, 96 times 0, 1, 185 times 0) [i] based on linear OA(3239, 1027, F32, 20) (dual of [1027, 988, 21]-code), using
(27, 27+20, 1733078)-Net in Base 32 — Upper bound on s
There is no (27, 47, 1733079)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 55214 048089 290648 753842 549593 000678 609091 126744 933554 891720 099621 535294 > 3247 [i]