Best Known (69, 69+17, s)-Nets in Base 4
(69, 69+17, 2048)-Net over F4 — Constructive and digital
Digital (69, 86, 2048)-net over F4, using
- 41 times duplication [i] based on digital (68, 85, 2048)-net over F4, using
- net defined by OOA [i] based on linear OOA(485, 2048, F4, 17, 17) (dual of [(2048, 17), 34731, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(485, 16385, F4, 17) (dual of [16385, 16300, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(485, 16385, F4, 17) (dual of [16385, 16300, 18]-code), using
- net defined by OOA [i] based on linear OOA(485, 2048, F4, 17, 17) (dual of [(2048, 17), 34731, 18]-NRT-code), using
(69, 69+17, 8196)-Net over F4 — Digital
Digital (69, 86, 8196)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(486, 8196, F4, 2, 17) (dual of [(8196, 2), 16306, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(486, 16392, F4, 17) (dual of [16392, 16306, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(478, 16384, F4, 15) (dual of [16384, 16306, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(41, 8, F4, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- OOA 2-folding [i] based on linear OA(486, 16392, F4, 17) (dual of [16392, 16306, 18]-code), using
(69, 69+17, 3129358)-Net in Base 4 — Upper bound on s
There is no (69, 86, 3129359)-net in base 4, because
- 1 times m-reduction [i] would yield (69, 85, 3129359)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1496 578379 641307 541561 935368 018452 119011 761538 259419 > 485 [i]