Best Known (81, 81+25, s)-Nets in Base 32
(81, 81+25, 87384)-Net over F32 — Constructive and digital
Digital (81, 106, 87384)-net over F32, using
- 323 times duplication [i] based on digital (78, 103, 87384)-net over F32, using
- net defined by OOA [i] based on linear OOA(32103, 87384, F32, 25, 25) (dual of [(87384, 25), 2184497, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(32103, 1048609, F32, 25) (dual of [1048609, 1048506, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(3297, 1048576, F32, 25) (dual of [1048576, 1048479, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3269, 1048576, F32, 18) (dual of [1048576, 1048507, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(326, 33, F32, 6) (dual of [33, 27, 7]-code or 33-arc in PG(5,32)), using
- extended Reed–Solomon code RSe(27,32) [i]
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- OOA 12-folding and stacking with additional row [i] based on linear OA(32103, 1048609, F32, 25) (dual of [1048609, 1048506, 26]-code), using
- net defined by OOA [i] based on linear OOA(32103, 87384, F32, 25, 25) (dual of [(87384, 25), 2184497, 26]-NRT-code), using
(81, 81+25, 174763)-Net in Base 32 — Constructive
(81, 106, 174763)-net in base 32, using
- 321 times duplication [i] based on (80, 105, 174763)-net in base 32, using
- base change [i] based on digital (50, 75, 174763)-net over F128, using
- 1281 times duplication [i] based on digital (49, 74, 174763)-net over F128, using
- net defined by OOA [i] based on linear OOA(12874, 174763, F128, 25, 25) (dual of [(174763, 25), 4369001, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12874, 2097157, F128, 25) (dual of [2097157, 2097083, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(12874, 2097160, F128, 25) (dual of [2097160, 2097086, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(12873, 2097153, F128, 25) (dual of [2097153, 2097080, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(12867, 2097153, F128, 23) (dual of [2097153, 2097086, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12874, 2097160, F128, 25) (dual of [2097160, 2097086, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12874, 2097157, F128, 25) (dual of [2097157, 2097083, 26]-code), using
- net defined by OOA [i] based on linear OOA(12874, 174763, F128, 25, 25) (dual of [(174763, 25), 4369001, 26]-NRT-code), using
- 1281 times duplication [i] based on digital (49, 74, 174763)-net over F128, using
- base change [i] based on digital (50, 75, 174763)-net over F128, using
(81, 81+25, 1405240)-Net over F32 — Digital
Digital (81, 106, 1405240)-net over F32, using
(81, 81+25, large)-Net in Base 32 — Upper bound on s
There is no (81, 106, large)-net in base 32, because
- 23 times m-reduction [i] would yield (81, 83, large)-net in base 32, but