Best Known (60, 60+12, s)-Nets in Base 8
(60, 60+12, 349528)-Net over F8 — Constructive and digital
Digital (60, 72, 349528)-net over F8, using
- net defined by OOA [i] based on linear OOA(872, 349528, F8, 12, 12) (dual of [(349528, 12), 4194264, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(872, 2097168, F8, 12) (dual of [2097168, 2097096, 13]-code), using
- construction X4 applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(871, 2097152, F8, 12) (dual of [2097152, 2097081, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(857, 2097152, F8, 10) (dual of [2097152, 2097095, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(815, 16, F8, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,8)), using
- dual of repetition code with length 16 [i]
- linear OA(81, 16, F8, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- construction X4 applied to Ce(11) ⊂ Ce(9) [i] based on
- OA 6-folding and stacking [i] based on linear OA(872, 2097168, F8, 12) (dual of [2097168, 2097096, 13]-code), using
(60, 60+12, 1670382)-Net over F8 — Digital
Digital (60, 72, 1670382)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(872, 1670382, F8, 12) (dual of [1670382, 1670310, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(872, 2097167, F8, 12) (dual of [2097167, 2097095, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(871, 2097152, F8, 12) (dual of [2097152, 2097081, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(857, 2097152, F8, 10) (dual of [2097152, 2097095, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(81, 15, F8, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(872, 2097167, F8, 12) (dual of [2097167, 2097095, 13]-code), using
(60, 60+12, large)-Net in Base 8 — Upper bound on s
There is no (60, 72, large)-net in base 8, because
- 10 times m-reduction [i] would yield (60, 62, large)-net in base 8, but