Best Known (4, 4+4, s)-Nets in Base 128
(4, 4+4, 16512)-Net over F128 — Constructive and digital
Digital (4, 8, 16512)-net over F128, using
- net defined by OOA [i] based on linear OOA(1288, 16512, F128, 4, 4) (dual of [(16512, 4), 66040, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(1288, 16512, F128, 3, 4) (dual of [(16512, 3), 49528, 5]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1280, s, F128, 3, 0) with arbitrarily large s, using
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code) (see above)
- linear OOA(1281, 129, F128, 3, 1) (dual of [(129, 3), 386, 2]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1281, s, F128, 3, 1) with arbitrarily large s, using
- appending 2 arbitrary columns [i] based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- discarding factors / shortening the dual code based on linear OOA(1281, s, F128, 3, 1) with arbitrarily large s, using
- linear OOA(1281, 129, F128, 3, 1) (dual of [(129, 3), 386, 2]-NRT-code) (see above)
- linear OOA(1282, 129, F128, 3, 2) (dual of [(129, 3), 385, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;385,128) [i]
- linear OOA(1284, 129, F128, 3, 4) (dual of [(129, 3), 383, 5]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;383,128) [i]
- linear OOA(1280, 129, F128, 3, 0) (dual of [(129, 3), 387, 1]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(1288, 16512, F128, 3, 4) (dual of [(16512, 3), 49528, 5]-NRT-code), using
(4, 4+4, 32769)-Net in Base 128 — Constructive
(4, 8, 32769)-net in base 128, using
- base change [i] based on digital (3, 7, 32769)-net over F256, using
- net defined by OOA [i] based on linear OOA(2567, 32769, F256, 4, 4) (dual of [(32769, 4), 131069, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2567, 65538, F256, 4) (dual of [65538, 65531, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(2567, 65536, F256, 4) (dual of [65536, 65529, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(2565, 65536, F256, 3) (dual of [65536, 65531, 4]-code or 65536-cap in PG(4,256)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(2567, 65538, F256, 4) (dual of [65538, 65531, 5]-code), using
- net defined by OOA [i] based on linear OOA(2567, 32769, F256, 4, 4) (dual of [(32769, 4), 131069, 5]-NRT-code), using
(4, 4+4, 65538)-Net in Base 128
(4, 8, 65538)-net in base 128, using
- base change [i] based on digital (3, 7, 65538)-net over F256, using
- net defined by OOA [i] based on linear OOA(2567, 65538, F256, 4, 4) (dual of [(65538, 4), 262145, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(2567, 65538, F256, 3, 4) (dual of [(65538, 3), 196607, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2567, 65538, F256, 4) (dual of [65538, 65531, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(2567, 65536, F256, 4) (dual of [65536, 65529, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(2565, 65536, F256, 3) (dual of [65536, 65531, 4]-code or 65536-cap in PG(4,256)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2567, 65538, F256, 4) (dual of [65538, 65531, 5]-code), using
- appending kth column [i] based on linear OOA(2567, 65538, F256, 3, 4) (dual of [(65538, 3), 196607, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(2567, 65538, F256, 4, 4) (dual of [(65538, 4), 262145, 5]-NRT-code), using
(4, 4+4, 2989173)-Net in Base 128 — Upper bound on s
There is no (4, 8, 2989174)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 72057 632381 198722 > 1288 [i]