Best Known (89−15, 89, s)-Nets in Base 8
(89−15, 89, 74900)-Net over F8 — Constructive and digital
Digital (74, 89, 74900)-net over F8, using
- 81 times duplication [i] based on digital (73, 88, 74900)-net over F8, using
- net defined by OOA [i] based on linear OOA(888, 74900, F8, 15, 15) (dual of [(74900, 15), 1123412, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(888, 524301, F8, 15) (dual of [524301, 524213, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(888, 524304, F8, 15) (dual of [524304, 524216, 16]-code), using
- trace code [i] based on linear OA(6444, 262152, F64, 15) (dual of [262152, 262108, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(6443, 262145, F64, 15) (dual of [262145, 262102, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(6437, 262145, F64, 13) (dual of [262145, 262108, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 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
- trace code [i] based on linear OA(6444, 262152, F64, 15) (dual of [262152, 262108, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(888, 524304, F8, 15) (dual of [524304, 524216, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(888, 524301, F8, 15) (dual of [524301, 524213, 16]-code), using
- net defined by OOA [i] based on linear OOA(888, 74900, F8, 15, 15) (dual of [(74900, 15), 1123412, 16]-NRT-code), using
(89−15, 89, 524306)-Net over F8 — Digital
Digital (74, 89, 524306)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(889, 524306, F8, 15) (dual of [524306, 524217, 16]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(888, 524304, F8, 15) (dual of [524304, 524216, 16]-code), using
- trace code [i] based on linear OA(6444, 262152, F64, 15) (dual of [262152, 262108, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(6443, 262145, F64, 15) (dual of [262145, 262102, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(6437, 262145, F64, 13) (dual of [262145, 262108, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 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
- trace code [i] based on linear OA(6444, 262152, F64, 15) (dual of [262152, 262108, 16]-code), using
- linear OA(888, 524305, F8, 14) (dual of [524305, 524217, 15]-code), using Gilbert–Varšamov bound and bm = 888 > Vbs−1(k−1) = 3519 754360 311078 247598 767231 296367 212708 937478 219662 564852 787360 440423 975191 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(888, 524304, F8, 15) (dual of [524304, 524216, 16]-code), using
- construction X with Varšamov bound [i] based on
(89−15, 89, large)-Net in Base 8 — Upper bound on s
There is no (74, 89, large)-net in base 8, because
- 13 times m-reduction [i] would yield (74, 76, large)-net in base 8, but