Best Known (25, 25+13, s)-Nets in Base 8
(25, 25+13, 260)-Net over F8 — Constructive and digital
Digital (25, 38, 260)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (6, 12, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 6, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 6, 65)-net over F64, using
- digital (13, 26, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 13, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- trace code for nets [i] based on digital (0, 13, 65)-net over F64, using
- digital (6, 12, 130)-net over F8, using
(25, 25+13, 516)-Net in Base 8 — Constructive
(25, 38, 516)-net in base 8, using
- trace code for nets [i] based on (6, 19, 258)-net in base 64, using
- 1 times m-reduction [i] based on (6, 20, 258)-net in base 64, using
- base change [i] based on digital (1, 15, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- base change [i] based on digital (1, 15, 258)-net over F256, using
- 1 times m-reduction [i] based on (6, 20, 258)-net in base 64, using
(25, 25+13, 587)-Net over F8 — Digital
Digital (25, 38, 587)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(838, 587, F8, 13) (dual of [587, 549, 14]-code), using
- 66 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 17 times 0, 1, 42 times 0) [i] based on linear OA(834, 517, F8, 13) (dual of [517, 483, 14]-code), using
- construction XX applied to C1 = C([510,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([510,11]) [i] based on
- linear OA(831, 511, F8, 12) (dual of [511, 480, 13]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(831, 511, F8, 12) (dual of [511, 480, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(834, 511, F8, 13) (dual of [511, 477, 14]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(828, 511, F8, 11) (dual of [511, 483, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(80, 3, F8, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(80, 3, F8, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([510,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([510,11]) [i] based on
- 66 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 17 times 0, 1, 42 times 0) [i] based on linear OA(834, 517, F8, 13) (dual of [517, 483, 14]-code), using
(25, 25+13, 158551)-Net in Base 8 — Upper bound on s
There is no (25, 38, 158552)-net in base 8, because
- 1 times m-reduction [i] would yield (25, 37, 158552)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 2596 218644 889416 292803 969013 110275 > 837 [i]