Best Known (109, 109+21, s)-Nets in Base 4
(109, 109+21, 6557)-Net over F4 — Constructive and digital
Digital (109, 130, 6557)-net over F4, using
- net defined by OOA [i] based on linear OOA(4130, 6557, F4, 21, 21) (dual of [(6557, 21), 137567, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4130, 65571, F4, 21) (dual of [65571, 65441, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4130, 65572, F4, 21) (dual of [65572, 65442, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(49, 36, F4, 5) (dual of [36, 27, 6]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(4130, 65572, F4, 21) (dual of [65572, 65442, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4130, 65571, F4, 21) (dual of [65571, 65441, 22]-code), using
(109, 109+21, 32788)-Net over F4 — Digital
Digital (109, 130, 32788)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4130, 32788, F4, 2, 21) (dual of [(32788, 2), 65446, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4130, 65576, F4, 21) (dual of [65576, 65446, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4130, 65577, F4, 21) (dual of [65577, 65447, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(49, 41, F4, 5) (dual of [41, 32, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- a “DaH†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(4130, 65577, F4, 21) (dual of [65577, 65447, 22]-code), using
- OOA 2-folding [i] based on linear OA(4130, 65576, F4, 21) (dual of [65576, 65446, 22]-code), using
(109, 109+21, large)-Net in Base 4 — Upper bound on s
There is no (109, 130, large)-net in base 4, because
- 19 times m-reduction [i] would yield (109, 111, large)-net in base 4, but