We define a linear code Cη(δT , δX) by evaluating polynomials of bidegree (δT , δX) in the Cox ring on Fq-rational points of the Hirzebruch surface on the finite field Fq. We give explicit parameters of the code, notably using Gröbner bases. The minimum distance provides an upper bound of the number of Fq-rational points of a non-filling curve on a Hirzebruch surface. We also display some punct...

In this letter, we discussed some properties of characteristic generators for a finite Abelian group code, proved that any two characteristic generators can not start (end) at the same position and have the same order of the starting (ending) components simultaneously, and that the number of all characteristic generators can be directly computed from the group code itself. These properties are ...

The Assmus{Mattson theorem is a method to nd designs in linear codes over a nite eld. The purpose of this paper is to give an analogue of this theorem for Z 4 -codes by using the harmonic weight enumerator which is introduced by Bachoc. This theorem can nd some 5-designs in the lifted Golay code over Z 4 which were discovered previously by other methods. Index Terms { Z 4 -codes, Assmus{Mattson...

Let r, s, t be three positive integers and C be a binary linear code of lenght r + s + t. We say that C is a triple cyclic code of lenght (r, s, t) over Z 2 if the set of coordinates can be partitioned into three parts that any cyclic shift of the coordinates of the parts leaves invariant the code. These codes can be considered as Z 2 [x]-submodules of Z2[x] x r −1 × Z2[x] x s −1 × Z2[x] x t −1...

and α0, . . . , αn−1 are distinct elements in Fq. Typically, q = n and the αi’a are all thje elements of Fq, or n = q − 1 and the αi’s are all the nonzero elements of of Fq. The linearity of an RS code C can be easily verified by checking the conditions for closure under addition and scalar multiplication. Let c, c′ ∈ C be codewords corresponding to the messages m = (m0, . . . ,mk−1) and m′ = (...

We show that the Feng-Rao bound for dual codes and a similar bound by Andersen and Geil [1] for primary codes are consequences of each other. This implies that the Feng-Rao decoding algorithm can be applied to decode primary codes up to half their designed minimum distance. The technique applies to any linear code for which information on well-behaving pairs is available. Consequently we are ab...

The binomial moments of a linear code are synonymously related to its weight distribution and appear naturally in studies involving the weight distribution of the code. In particular , the binomial moments have been used for establishing bounds on the undetected error probability and, though implicitly, for the study of good and proper codes. In this work we sharpen known bounds on the binomial...

It is known that a linear two-weight code C over a finite field Fq corresponds both to a multiset in a projective space over Fq that meets every hyperplane in either a or b points for some integers a < b , and to a strongly regular graph whose vertices may be identified with the codewords of C . Here we extend this classical result to the case of a ring-linear code with exactly two nonzero homo...

For an [n, k, d]3 code C with gcd(d, 3) = 1, we define a map wG from Σ = PG(k − 1, 3) to the set of weights of codewords of C through a generator matrix G. A t-flat Π in Σ is called an (i, j)t flat if (i, j) = (|Π ∩ F0|, |Π ∩ F1|), where F0 = {P ∈ Σ | wG(P ) ≡ 0 (mod 3)}, F1 = {P ∈ Σ | wG(P ) 6≡ 0, d (mod 3)}. We give geometric characterizations of (i, j)t flats, which involve quadrics. As an a...

We apply the well-known operator of sequences, the derivative D, on codewords of linear codes. The depth of a codeword c is the smallest integer i such that Dc (the derivative applied i consecutive times) is zero. We show that the depth distribution of the nonzero codewords of an [n; k] linear code consists of exactly k nonzero values, and its generator matrix can be constructed from any k nonz...