Best Known (64, 64+15, s)-Nets in Base 4
(64, 64+15, 2342)-Net over F4 — Constructive and digital
Digital (64, 79, 2342)-net over F4, using
- net defined by OOA [i] based on linear OOA(479, 2342, F4, 15, 15) (dual of [(2342, 15), 35051, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(479, 16395, F4, 15) (dual of [16395, 16316, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(479, 16399, F4, 15) (dual of [16399, 16320, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- 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(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(41, 15, F4, 1) (dual of [15, 14, 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(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(479, 16399, F4, 15) (dual of [16399, 16320, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(479, 16395, F4, 15) (dual of [16395, 16316, 16]-code), using
(64, 64+15, 8200)-Net over F4 — Digital
Digital (64, 79, 8200)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(479, 8200, F4, 2, 15) (dual of [(8200, 2), 16321, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(479, 16400, F4, 15) (dual of [16400, 16321, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- 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(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(415, 16, F4, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,4)), using
- dual of repetition code with length 16 [i]
- linear OA(41, 16, F4, 1) (dual of [16, 15, 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 X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(479, 16400, F4, 15) (dual of [16400, 16321, 16]-code), using
(64, 64+15, 5760570)-Net in Base 4 — Upper bound on s
There is no (64, 79, 5760571)-net in base 4, because
- 1 times m-reduction [i] would yield (64, 78, 5760571)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 91343 951990 878670 290142 620920 478603 751821 125376 > 478 [i]