Best Known (81, 81+16, s)-Nets in Base 4
(81, 81+16, 8193)-Net over F4 — Constructive and digital
Digital (81, 97, 8193)-net over F4, using
- net defined by OOA [i] based on linear OOA(497, 8193, F4, 16, 16) (dual of [(8193, 16), 130991, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(497, 65544, F4, 16) (dual of [65544, 65447, 17]-code), using
- 1 times truncation [i] based on linear OA(498, 65545, F4, 17) (dual of [65545, 65447, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 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
- 1 times truncation [i] based on linear OA(498, 65545, F4, 17) (dual of [65545, 65447, 18]-code), using
- OA 8-folding and stacking [i] based on linear OA(497, 65544, F4, 16) (dual of [65544, 65447, 17]-code), using
(81, 81+16, 32772)-Net over F4 — Digital
Digital (81, 97, 32772)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(497, 32772, F4, 2, 16) (dual of [(32772, 2), 65447, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(497, 65544, F4, 16) (dual of [65544, 65447, 17]-code), using
- 1 times truncation [i] based on linear OA(498, 65545, F4, 17) (dual of [65545, 65447, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 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
- 1 times truncation [i] based on linear OA(498, 65545, F4, 17) (dual of [65545, 65447, 18]-code), using
- OOA 2-folding [i] based on linear OA(497, 65544, F4, 16) (dual of [65544, 65447, 17]-code), using
(81, 81+16, large)-Net in Base 4 — Upper bound on s
There is no (81, 97, large)-net in base 4, because
- 14 times m-reduction [i] would yield (81, 83, large)-net in base 4, but