Best Known (41−13, 41, s)-Nets in Base 32
(41−13, 41, 5464)-Net over F32 — Constructive and digital
Digital (28, 41, 5464)-net over F32, using
- net defined by OOA [i] based on linear OOA(3241, 5464, F32, 13, 13) (dual of [(5464, 13), 70991, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3241, 32785, F32, 13) (dual of [32785, 32744, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3241, 32787, F32, 13) (dual of [32787, 32746, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(3237, 32768, F32, 13) (dual of [32768, 32731, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(3222, 32768, F32, 8) (dual of [32768, 32746, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(324, 19, F32, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,32)), using
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- Reed–Solomon code RS(28,32) [i]
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(3241, 32787, F32, 13) (dual of [32787, 32746, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3241, 32785, F32, 13) (dual of [32785, 32744, 14]-code), using
(41−13, 41, 10923)-Net in Base 32 — Constructive
(28, 41, 10923)-net in base 32, using
- net defined by OOA [i] based on OOA(3241, 10923, S32, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3241, 65539, S32, 13), using
- 1 times code embedding in larger space [i] based on OA(3240, 65538, S32, 13), using
- discarding parts of the base [i] based on linear OA(25625, 65538, F256, 13) (dual of [65538, 65513, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(25625, 65536, F256, 13) (dual of [65536, 65511, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(25625, 65538, F256, 13) (dual of [65538, 65513, 14]-code), using
- 1 times code embedding in larger space [i] based on OA(3240, 65538, S32, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3241, 65539, S32, 13), using
(41−13, 41, 32787)-Net over F32 — Digital
Digital (28, 41, 32787)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3241, 32787, F32, 13) (dual of [32787, 32746, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(3237, 32768, F32, 13) (dual of [32768, 32731, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(3222, 32768, F32, 8) (dual of [32768, 32746, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(324, 19, F32, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,32)), using
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- Reed–Solomon code RS(28,32) [i]
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
(41−13, 41, large)-Net in Base 32 — Upper bound on s
There is no (28, 41, large)-net in base 32, because
- 11 times m-reduction [i] would yield (28, 30, large)-net in base 32, but