Best Known (30−15, 30, s)-Nets in Base 32
(30−15, 30, 147)-Net over F32 — Constructive and digital
Digital (15, 30, 147)-net over F32, using
- net defined by OOA [i] based on linear OOA(3230, 147, F32, 15, 15) (dual of [(147, 15), 2175, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3230, 1030, F32, 15) (dual of [1030, 1000, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(3229, 1025, F32, 15) (dual of [1025, 996, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(3225, 1025, F32, 13) (dual of [1025, 1000, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(321, 5, F32, 1) (dual of [5, 4, 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([0,7]) ⊂ C([0,6]) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(3230, 1030, F32, 15) (dual of [1030, 1000, 16]-code), using
(30−15, 30, 260)-Net in Base 32 — Constructive
(15, 30, 260)-net in base 32, using
- 2 times m-reduction [i] based on (15, 32, 260)-net in base 32, using
- base change [i] based on digital (3, 20, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
- base change [i] based on digital (3, 20, 260)-net over F256, using
(30−15, 30, 515)-Net over F32 — Digital
Digital (15, 30, 515)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3230, 515, F32, 2, 15) (dual of [(515, 2), 1000, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3230, 1030, F32, 15) (dual of [1030, 1000, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(3229, 1025, F32, 15) (dual of [1025, 996, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(3225, 1025, F32, 13) (dual of [1025, 1000, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(321, 5, F32, 1) (dual of [5, 4, 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([0,7]) ⊂ C([0,6]) [i] based on
- OOA 2-folding [i] based on linear OA(3230, 1030, F32, 15) (dual of [1030, 1000, 16]-code), using
(30−15, 30, 187573)-Net in Base 32 — Upper bound on s
There is no (15, 30, 187574)-net in base 32, because
- 1 times m-reduction [i] would yield (15, 29, 187574)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 44 602795 569722 765998 002678 385259 251880 157144 > 3229 [i]