Best Known (33−11, 33, s)-Nets in Base 32
(33−11, 33, 6555)-Net over F32 — Constructive and digital
Digital (22, 33, 6555)-net over F32, using
- 321 times duplication [i] based on digital (21, 32, 6555)-net over F32, using
- net defined by OOA [i] based on linear OOA(3232, 6555, F32, 11, 11) (dual of [(6555, 11), 72073, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3232, 32776, F32, 11) (dual of [32776, 32744, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(3231, 32769, F32, 11) (dual of [32769, 32738, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(3225, 32769, F32, 9) (dual of [32769, 32744, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(321, 7, F32, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 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
- OOA 5-folding and stacking with additional row [i] based on linear OA(3232, 32776, F32, 11) (dual of [32776, 32744, 12]-code), using
- net defined by OOA [i] based on linear OOA(3232, 6555, F32, 11, 11) (dual of [(6555, 11), 72073, 12]-NRT-code), using
(33−11, 33, 30059)-Net over F32 — Digital
Digital (22, 33, 30059)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3233, 30059, F32, 11) (dual of [30059, 30026, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3233, 32779, F32, 11) (dual of [32779, 32746, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(3231, 32768, F32, 11) (dual of [32768, 32737, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(322, 11, F32, 2) (dual of [11, 9, 3]-code or 11-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
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(3233, 32779, F32, 11) (dual of [32779, 32746, 12]-code), using
(33−11, 33, large)-Net in Base 32 — Upper bound on s
There is no (22, 33, large)-net in base 32, because
- 9 times m-reduction [i] would yield (22, 24, large)-net in base 32, but