Best Known (25, 25+9, s)-Nets in Base 8
(25, 25+9, 2048)-Net over F8 — Constructive and digital
Digital (25, 34, 2048)-net over F8, using
- net defined by OOA [i] based on linear OOA(834, 2048, F8, 9, 9) (dual of [(2048, 9), 18398, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(834, 8193, F8, 9) (dual of [8193, 8159, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(834, 8194, F8, 9) (dual of [8194, 8160, 10]-code), using
- trace code [i] based on linear OA(6417, 4097, F64, 9) (dual of [4097, 4080, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- trace code [i] based on linear OA(6417, 4097, F64, 9) (dual of [4097, 4080, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(834, 8194, F8, 9) (dual of [8194, 8160, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(834, 8193, F8, 9) (dual of [8193, 8159, 10]-code), using
(25, 25+9, 8196)-Net over F8 — Digital
Digital (25, 34, 8196)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(834, 8196, F8, 9) (dual of [8196, 8162, 10]-code), using
- trace code [i] based on linear OA(6417, 4098, F64, 9) (dual of [4098, 4081, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(6417, 4096, F64, 9) (dual of [4096, 4079, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- trace code [i] based on linear OA(6417, 4098, F64, 9) (dual of [4098, 4081, 10]-code), using
(25, 25+9, large)-Net in Base 8 — Upper bound on s
There is no (25, 34, large)-net in base 8, because
- 7 times m-reduction [i] would yield (25, 27, large)-net in base 8, but