lower bound: | 70 |
upper bound: | 76 |
Construction of a linear code [172,17,70] over GF(2): [1]: [24, 12, 8] Linear Code over GF(2) Extend the QRCode over GF(2)of length 23 [2]: [20, 8, 8] Linear Code over GF(2) Shortening of [1] at { 1, 2, 3, 4 } [3]: [153, 10, 70] Quasicyclic of degree 3 Linear Code over GF(2) QuasiCyclicCode of length 153 with generating polynomials: x^50 + x^48 + x^44 + x^43 + x^39 + x^37 + x^36 + x^35 + x^34 + x^31 + x^26 + x^22 + x^20 + x^19 + x^18 + x^16 + x^10 + 1, x^46 + x^45 + x^44 + x^43 + x^41 + x^37 + x^34 + x^33 + x^30 + x^28 + x^23 + x^22 + x^19 + x^18 + x^16 + x^13 + x^12 + x^10 + x^9 + x^7 + x^6 + x^5 + x^3 + x^2 + x + 1, x^50 + x^49 + x^48 + x^47 + x^44 + x^43 + x^38 + x^36 + x^35 + x^32 + x^30 + x^29 + x^27 + x^26 + x^24 + x^22 + x^21 + x^18 + x^17 + x^16 + x^14 + x^12 + x^7 + x^5 + x^2 + 1 [4]: [153, 18, 62] Quasicyclic of degree 3 Linear Code over GF(2) QuasiCyclicCode of length 153 with generating polynomials: x^48 + x^46 + x^44 + x^40 + x^36 + x^35 + x^34 + x^31 + x^29 + x^27 + x^23 + x^19 + x^18 + 1, x^50 + x^49 + x^46 + x^45 + x^44 + x^42 + x^41 + x^39 + x^38 + x^36 + x^34 + x^33 + x^31 + x^30 + x^27 + x^26 + x^19 + x^18 + x^17 + x^15 + x^14 + x^13 + x^12 + x^11 + x^9 + x^8 + x^7 + x^5 + x^4 + x, x^49 + x^45 + x^44 + x^42 + x^41 + x^40 + x^39 + x^38 + x^37 + x^35 + x^33 + x^27 + x^26 + x^24 + x^22 + x^20 + x^17 + x^16 + x^15 + x^11 + x^9 + x^8 + x^6 + x^4 + x + 1 [5]: [173, 18, 70] Linear Code over GF(2) ConstructionX using [4] [3] and [2] [6]: [172, 17, 70] Linear Code over GF(2) Shortening of [5] at { 173 } last modified: 2003-04-09
Lb(172,17) = 70 is found by shortening of: Lb(173,18) = 70 GW2 Ub(172,17) = 76 otherwise adding a parity check bit would contradict: Ub(173,17) = 77 BK
GW2: M. Grassl & G. White, New Codes from Chains of Quasi-cyclic Codes, ISIT 2005.
Notes
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