Best Known (13, 22, s)-Nets in Base 32
(13, 22, 300)-Net over F32 — Constructive and digital
Digital (13, 22, 300)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 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 (8, 17, 256)-net over F32, using
- net defined by OOA [i] based on linear OOA(3217, 256, F32, 9, 9) (dual of [(256, 9), 2287, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on 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]
- OOA 4-folding and stacking with additional row [i] based on linear OA(3217, 1025, F32, 9) (dual of [1025, 1008, 10]-code), using
- net defined by OOA [i] based on linear OOA(3217, 256, F32, 9, 9) (dual of [(256, 9), 2287, 10]-NRT-code), using
- digital (1, 5, 44)-net over F32, using
(13, 22, 1025)-Net in Base 32 — Constructive
(13, 22, 1025)-net in base 32, using
- net defined by OOA [i] based on OOA(3222, 1025, S32, 9, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(3222, 4101, S32, 9), using
- discarding factors based on OA(3222, 4102, S32, 9), using
- discarding parts of the base [i] based on linear OA(6418, 4102, F64, 9) (dual of [4102, 4084, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- 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(6413, 4097, F64, 7) (dual of [4097, 4084, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,4]) ⊂ C([0,3]) [i] based on
- discarding parts of the base [i] based on linear OA(6418, 4102, F64, 9) (dual of [4102, 4084, 10]-code), using
- discarding factors based on OA(3222, 4102, S32, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(3222, 4101, S32, 9), using
(13, 22, 1701)-Net over F32 — Digital
Digital (13, 22, 1701)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3222, 1701, F32, 9) (dual of [1701, 1679, 10]-code), using
- 669 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 32 times 0, 1, 156 times 0, 1, 472 times 0) [i] based on linear OA(3217, 1027, F32, 9) (dual of [1027, 1010, 10]-code), using
- construction XX applied to C1 = C([1022,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([1022,7]) [i] based on
- linear OA(3215, 1023, F32, 8) (dual of [1023, 1008, 9]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(3215, 1023, F32, 8) (dual of [1023, 1008, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(3217, 1023, F32, 9) (dual of [1023, 1006, 10]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,7}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(3213, 1023, F32, 7) (dual of [1023, 1010, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [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([1022,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([1022,7]) [i] based on
- 669 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 32 times 0, 1, 156 times 0, 1, 472 times 0) [i] based on linear OA(3217, 1027, F32, 9) (dual of [1027, 1010, 10]-code), using
(13, 22, 5698077)-Net in Base 32 — Upper bound on s
There is no (13, 22, 5698078)-net in base 32, because
- 1 times m-reduction [i] would yield (13, 21, 5698078)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 40 564823 434327 897939 962337 772999 > 3221 [i]