Best Known (16, 27, s)-Nets in Base 32
(16, 27, 297)-Net over F32 — Constructive and digital
Digital (16, 27, 297)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 33)-net over F32, using
- s-reduction based on digital (0, 1, s)-net over F32 with arbitrarily large s, using
- digital (0, 1, 33)-net over F32 (see above)
- digital (0, 1, 33)-net over F32 (see above)
- digital (0, 1, 33)-net over F32 (see above)
- digital (0, 2, 33)-net over F32, using
- digital (0, 2, 33)-net over F32 (see above)
- digital (0, 3, 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
- digital (0, 5, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 11, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 1, 33)-net over F32, using
(16, 27, 820)-Net in Base 32 — Constructive
(16, 27, 820)-net in base 32, using
- net defined by OOA [i] based on OOA(3227, 820, S32, 11, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(3227, 4101, S32, 11), using
- discarding factors based on OA(3227, 4102, S32, 11), using
- discarding parts of the base [i] based on linear OA(6422, 4102, F64, 11) (dual of [4102, 4080, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(6421, 4097, F64, 11) (dual of [4097, 4076, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(6417, 4097, F64, 9) (dual of [4097, 4080, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- discarding parts of the base [i] based on linear OA(6422, 4102, F64, 11) (dual of [4102, 4080, 12]-code), using
- discarding factors based on OA(3227, 4102, S32, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(3227, 4101, S32, 11), using
(16, 27, 1706)-Net over F32 — Digital
Digital (16, 27, 1706)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3227, 1706, F32, 11) (dual of [1706, 1679, 12]-code), using
- 673 step Varšamov–Edel lengthening with (ri) = (2, 1, 6 times 0, 1, 43 times 0, 1, 182 times 0, 1, 437 times 0) [i] based on linear OA(3221, 1027, F32, 11) (dual of [1027, 1006, 12]-code), using
- construction XX applied to C1 = C([1022,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([1022,9]) [i] based on
- linear OA(3219, 1023, F32, 10) (dual of [1023, 1004, 11]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,8}, and designed minimum distance d ≥ |I|+1 = 11 [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(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(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(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([1022,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([1022,9]) [i] based on
- 673 step Varšamov–Edel lengthening with (ri) = (2, 1, 6 times 0, 1, 43 times 0, 1, 182 times 0, 1, 437 times 0) [i] based on linear OA(3221, 1027, F32, 11) (dual of [1027, 1006, 12]-code), using
(16, 27, 5639677)-Net in Base 32 — Upper bound on s
There is no (16, 27, 5639678)-net in base 32, because
- 1 times m-reduction [i] would yield (16, 26, 5639678)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 1361 129972 970941 463709 648249 732671 590357 > 3226 [i]