Best Known (11, 20, s)-Nets in Base 32
(11, 20, 258)-Net over F32 — Constructive and digital
Digital (11, 20, 258)-net over F32, using
- 321 times duplication [i] based on digital (10, 19, 258)-net over F32, using
- net defined by OOA [i] based on linear OOA(3219, 258, F32, 9, 9) (dual of [(258, 9), 2303, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(3219, 1033, F32, 9) (dual of [1033, 1014, 10]-code), using
- construction XX applied to C1 = C([1020,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1020,5]) [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 = {−3,−2,…,4}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(3211, 1023, F32, 6) (dual of [1023, 1012, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [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 = {−3,−2,…,5}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(329, 1023, F32, 5) (dual of [1023, 1014, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [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([1020,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1020,5]) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(3219, 1033, F32, 9) (dual of [1033, 1014, 10]-code), using
- net defined by OOA [i] based on linear OOA(3219, 258, F32, 9, 9) (dual of [(258, 9), 2303, 10]-NRT-code), using
(11, 20, 322)-Net in Base 32 — Constructive
(11, 20, 322)-net in base 32, using
- (u, u+v)-construction [i] based on
- (1, 5, 65)-net in base 32, using
- 1 times m-reduction [i] based on (1, 6, 65)-net in base 32, using
- base change [i] based on digital (0, 5, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- base change [i] based on digital (0, 5, 65)-net over F64, using
- 1 times m-reduction [i] based on (1, 6, 65)-net in base 32, using
- (6, 15, 257)-net in base 32, using
- 1 times m-reduction [i] based on (6, 16, 257)-net in base 32, using
- base change [i] based on digital (0, 10, 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, 10, 257)-net over F256, using
- 1 times m-reduction [i] based on (6, 16, 257)-net in base 32, using
- (1, 5, 65)-net in base 32, using
(11, 20, 1069)-Net over F32 — Digital
Digital (11, 20, 1069)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3220, 1069, F32, 9) (dual of [1069, 1049, 10]-code), using
- 39 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 32 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
- 39 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 32 times 0) [i] based on linear OA(3217, 1027, F32, 9) (dual of [1027, 1010, 10]-code), using
(11, 20, 1007286)-Net in Base 32 — Upper bound on s
There is no (11, 20, 1007287)-net in base 32, because
- 1 times m-reduction [i] would yield (11, 19, 1007287)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 39614 233677 270712 084404 147323 > 3219 [i]