Best Known (20−5, 20, s)-Nets in Base 32
(20−5, 20, 525346)-Net over F32 — Constructive and digital
Digital (15, 20, 525346)-net over F32, using
- net defined by OOA [i] based on linear OOA(3220, 525346, F32, 6, 5) (dual of [(525346, 6), 3152056, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(3220, 525347, F32, 2, 5) (dual of [(525347, 2), 1050674, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(323, 1057, F32, 2, 2) (dual of [(1057, 2), 2111, 3]-NRT-code), using
- appending kth column [i] based on linear OA(323, 1057, F32, 2) (dual of [1057, 1054, 3]-code), using
- Hamming code H(3,32) [i]
- appending kth column [i] based on linear OA(323, 1057, F32, 2) (dual of [1057, 1054, 3]-code), using
- linear OOA(3217, 524290, F32, 2, 5) (dual of [(524290, 2), 1048563, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3217, 1048580, F32, 5) (dual of [1048580, 1048563, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(3217, 1048576, F32, 5) (dual of [1048576, 1048559, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(3213, 1048576, F32, 4) (dual of [1048576, 1048563, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(320, 4, F32, 0) (dual of [4, 4, 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 X applied to Ce(4) ⊂ Ce(3) [i] based on
- OOA 2-folding [i] based on linear OA(3217, 1048580, F32, 5) (dual of [1048580, 1048563, 6]-code), using
- linear OOA(323, 1057, F32, 2, 2) (dual of [(1057, 2), 2111, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(3220, 525347, F32, 2, 5) (dual of [(525347, 2), 1050674, 6]-NRT-code), using
(20−5, 20, 1048580)-Net in Base 32 — Constructive
(15, 20, 1048580)-net in base 32, using
- net defined by OOA [i] based on OOA(3220, 1048580, S32, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(3220, 2097161, S32, 5), using
- discarding parts of the base [i] based on linear OA(12814, 2097161, F128, 5) (dual of [2097161, 2097147, 6]-code), using
- construction X4 applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(12813, 2097153, F128, 5) (dual of [2097153, 2097140, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(1287, 2097153, F128, 3) (dual of [2097153, 2097146, 4]-code or 2097153-cap in PG(6,128)), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(1287, 8, F128, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,128)), using
- dual of repetition code with length 8 [i]
- linear OA(1281, 8, F128, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, 128, F128, 1) (dual of [128, 127, 2]-code), using
- Reed–Solomon code RS(127,128) [i]
- discarding factors / shortening the dual code based on linear OA(1281, 128, F128, 1) (dual of [128, 127, 2]-code), using
- construction X4 applied to C([0,2]) ⊂ C([0,1]) [i] based on
- discarding parts of the base [i] based on linear OA(12814, 2097161, F128, 5) (dual of [2097161, 2097147, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on OA(3220, 2097161, S32, 5), using
(20−5, 20, 2395749)-Net over F32 — Digital
Digital (15, 20, 2395749)-net over F32, using
(20−5, 20, large)-Net in Base 32 — Upper bound on s
There is no (15, 20, large)-net in base 32, because
- 3 times m-reduction [i] would yield (15, 17, large)-net in base 32, but