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Algebraic geometric codes on minimal Hirzebruch surfaces. (English) Zbl 1433.14020

Fix a non-negative integer \(\eta \geq 0\),
Let \(H_\eta\) denote a Hirzebruch surface over a finite field \(k=GF(q)\), associated to the fan \(\Sigma_\eta\) spanned by \(\{(1,0),(0,1),(-1,0),(-\eta,-1)\}\).
A monomial \(M=T_1^{c_1}T_2^{c_2}X_1^{d_1}X_2^{d_2}\) of \(R=k[T_1,T_2,X_1,X_2]\) is said to have bidegree \((\delta_T,\delta_X)\) if \(\delta_T=c_1+c_2-\eta d_1,\delta_X=d_1+d_2.\)
The linear code \(C=C_\eta(\delta_T,\delta_X)\) is the evaluation code in the \(k\)-rational points of \(H_\eta\) in the ring \(R(\delta_T,\delta_X)\) of homogeneous polynomials of bidegree \((\delta_T,\delta_X)\). The author determines the dimension of \(C\) as a complicated combinatorial quantity (at least, too complicated to state here), but one which is straightforward to compute using a computer algebra system. In addition, the author finds a lower bound for the minimum distance in terms of certain Gröbner basis computations (again, too complicated to state here). In particular, Riemann-Roch-type considerations are avoided, but that’s balanced by the computational expense of computing a Gröbner basis.
What is most remarkable to this reviewer is that the author does not need to assume that the evaluation map defining \(C\) is injective. However, conditions on the injectivity are carefully explored.

MSC:

14G50 Applications to coding theory and cryptography of arithmetic geometry
94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
14G15 Finite ground fields in algebraic geometry
14M25 Toric varieties, Newton polyhedra, Okounkov bodies

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