Best Known (31−13, 31, s)-Nets in Base 128
(31−13, 31, 2859)-Net over F128 — Constructive and digital
Digital (18, 31, 2859)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- digital (12, 25, 2730)-net over F128, using
- net defined by OOA [i] based on linear OOA(12825, 2730, F128, 13, 13) (dual of [(2730, 13), 35465, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12825, 16381, F128, 13) (dual of [16381, 16356, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12825, 16381, F128, 13) (dual of [16381, 16356, 14]-code), using
- net defined by OOA [i] based on linear OOA(12825, 2730, F128, 13, 13) (dual of [(2730, 13), 35465, 14]-NRT-code), using
- digital (0, 6, 129)-net over F128, using
(31−13, 31, 10923)-Net in Base 128 — Constructive
(18, 31, 10923)-net in base 128, using
- 1281 times duplication [i] based on (17, 30, 10923)-net in base 128, using
- net defined by OOA [i] based on OOA(12830, 10923, S128, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(12830, 65539, S128, 13), using
- discarding factors based on OA(12830, 65542, S128, 13), using
- discarding parts of the base [i] based on linear OA(25626, 65542, F256, 13) (dual of [65542, 65516, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(25621, 65537, F256, 11) (dual of [65537, 65516, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- discarding parts of the base [i] based on linear OA(25626, 65542, F256, 13) (dual of [65542, 65516, 14]-code), using
- discarding factors based on OA(12830, 65542, S128, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(12830, 65539, S128, 13), using
- net defined by OOA [i] based on OOA(12830, 10923, S128, 13, 13), using
(31−13, 31, 16515)-Net over F128 — Digital
Digital (18, 31, 16515)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12831, 16515, F128, 13) (dual of [16515, 16484, 14]-code), using
- (u, u+v)-construction [i] based on
- linear OA(1286, 129, F128, 6) (dual of [129, 123, 7]-code or 129-arc in PG(5,128)), using
- extended Reed–Solomon code RSe(123,128) [i]
- linear OA(12825, 16386, F128, 13) (dual of [16386, 16361, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(1286, 129, F128, 6) (dual of [129, 123, 7]-code or 129-arc in PG(5,128)), using
- (u, u+v)-construction [i] based on
(31−13, 31, 18618)-Net in Base 128
(18, 31, 18618)-net in base 128, using
- 1 times m-reduction [i] based on (18, 32, 18618)-net in base 128, using
- base change [i] based on digital (14, 28, 18618)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25628, 18618, F256, 3, 14) (dual of [(18618, 3), 55826, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25628, 21847, F256, 3, 14) (dual of [(21847, 3), 65513, 15]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25628, 65541, F256, 14) (dual of [65541, 65513, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- OOA 3-folding [i] based on linear OA(25628, 65541, F256, 14) (dual of [65541, 65513, 15]-code), using
- discarding factors / shortening the dual code based on linear OOA(25628, 21847, F256, 3, 14) (dual of [(21847, 3), 65513, 15]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25628, 18618, F256, 3, 14) (dual of [(18618, 3), 55826, 15]-NRT-code), using
- base change [i] based on digital (14, 28, 18618)-net over F256, using
(31−13, 31, large)-Net in Base 128 — Upper bound on s
There is no (18, 31, large)-net in base 128, because
- 11 times m-reduction [i] would yield (18, 20, large)-net in base 128, but