Esempio n. 1
0
 def get_tau(s, l):
     "Return the decoding radius given this s and l"
     if s <= 0 or l <= 0:
         return -1
     return gilt(n - n / 2 * (s + 1) / (l + 1) - (k - 1) / 2 * l / s)
Esempio n. 2
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 def get_tau(s,l):
     "Return the decoding radius given this s and l"
     if s<=0 or l<=0:
         return -1
     return gilt(n - n/2*(s+1)/(l+1) - (k-1)/2*l/s)
Esempio n. 3
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    def guruswami_sudan_decoding_radius(C=None, n_k=None, l=None, s=None):
        r"""
        Returns the maximal decoding radius of the Guruswami-Sudan decoder and
        the parameter choices needed for this.

        If ``s`` is set but ``l`` is not it will return the best decoding radius using this ``s``
        alongside with the required ``l``. Vice versa for ``l``. If both are
        set, it returns the decoding radius given this parameter choice.

        INPUT:

        - ``C`` -- (default: ``None``) a :class:`GeneralizedReedSolomonCode`
        - ``n_k`` -- (default: ``None``) a pair of integers, respectively the
          length and the dimension of the :class:`GeneralizedReedSolomonCode`
        - ``s`` -- (default: ``None``) an integer, the multiplicity parameter of Guruswami-Sudan algorithm
        - ``l`` -- (default: ``None``) an integer, the list size parameter

        .. NOTE::

            One has to provide either ``C`` or ``n_k``. If none or both are
            given, an exception will be raised.

        OUTPUT:

        - ``(tau, (s, l))`` -- where
            - ``tau`` is the obtained decoding radius, and
            - ``s, ell`` are the multiplicity parameter, respectively list size
              parameter giving this radius.

        EXAMPLES::

            sage: n, k = 250, 70
            sage: codes.decoders.GRSGuruswamiSudanDecoder.guruswami_sudan_decoding_radius(n_k = (n, k))
            (118, (47, 89))

        One parameter can be restricted at a time::

            sage: n, k = 250, 70
            sage: codes.decoders.GRSGuruswamiSudanDecoder.guruswami_sudan_decoding_radius(n_k = (n, k), s=3)
            (109, (3, 5))
            sage: codes.decoders.GRSGuruswamiSudanDecoder.guruswami_sudan_decoding_radius(n_k = (n, k), l=7)
            (111, (4, 7))

        The function can also just compute the decoding radius given the parameters::

            sage: codes.decoders.GRSGuruswamiSudanDecoder.guruswami_sudan_decoding_radius(n_k = (n, k), s=2, l=6)
            (92, (2, 6))
        """
        n, k = n_k_params(C, n_k)

        def get_tau(s, l):
            "Return the decoding radius given this s and l"
            if s <= 0 or l <= 0:
                return -1
            return gilt(n - n / 2 * (s + 1) / (l + 1) - (k - 1) / 2 * l / s)

        if l is None and s is None:
            tau = gilt(johnson_radius(n, n - k + 1))
            return (tau,
                    GRSGuruswamiSudanDecoder.parameters_given_tau(tau,
                                                                  n_k=(n, k)))
        if l is not None and s is not None:
            return (get_tau(s, l), (s, l))

        # Either s or l is set, but not both. First a shared local function
        def find_integral_max(real_max, f):
            """Given a real (local) maximum of a function `f`, return that of
            the integers around `real_max` which gives the (local) integral
            maximum, and the value of at that point."""
            if real_max in ZZ:
                int_max = Integer(real_max)
                return (int_max, f(int_max))
            else:
                x_f = floor(real_max)
                x_c = x_f + 1
                f_f, f_c = f(x_f), f(x_c)
                return (x_f, f_f) if f_f >= f_c else (x_c, f_c)

        if s is not None:
            # maximising tau under condition
            # n*(s+1 choose 2) < (ell+1)*s*(n-tau) - (ell+1 choose 2)*(k-1)
            # knowing n and s, we can just minimise
            # ( n*(s+1 choose 2) + (ell+1 choose 2)*(k-1) )/(ell+1)
            # Differentiating and setting to zero yields ell best choice:
            lmax = sqrt(n * s * (s + 1.) / (k - 1.)) - 1.
            #the best integral value will be
            (l, tau) = find_integral_max(lmax, lambda l: get_tau(s, l))
            #Note that we have not proven that this ell is minimal in integral
            #sense! It just seems that this most often happens
            return (tau, (s, l))
        if l is not None:
            # Acquired similarly to when restricting s
            smax = sqrt((k - 1.) / n * l * (l + 1.))
            (s, tau) = find_integral_max(smax, lambda s: get_tau(s, l))
            return (tau, (s, l))
Esempio n. 4
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    def guruswami_sudan_decoding_radius(C = None, n_k = None, l = None, s = None):
        r"""
        Returns the maximal decoding radius of the Guruswami-Sudan decoder and
        the parameter choices needed for this.

        If ``s`` is set but ``l`` is not it will return the best decoding radius using this ``s``
        alongside with the required ``l``. Vice versa for ``l``. If both are
        set, it returns the decoding radius given this parameter choice.

        INPUT:

        - ``C`` -- (default: ``None``) a :class:`GeneralizedReedSolomonCode`
        - ``n_k`` -- (default: ``None``) a pair of integers, respectively the
          length and the dimension of the :class:`GeneralizedReedSolomonCode`
        - ``s`` -- (default: ``None``) an integer, the multiplicity parameter of Guruswami-Sudan algorithm
        - ``l`` -- (default: ``None``) an integer, the list size parameter

        .. NOTE::

            One has to provide either ``C`` or ``n_k``. If none or both are
            given, an exception will be raised.

        OUTPUT:

        - ``(tau, (s, l))`` -- where
            - ``tau`` is the obtained decoding radius, and
            - ``s, ell`` are the multiplicity parameter, respectively list size
              parameter giving this radius.

        EXAMPLES::

            sage: n, k = 250, 70
            sage: codes.decoders.GRSGuruswamiSudanDecoder.guruswami_sudan_decoding_radius(n_k = (n, k))
            (118, (47, 89))

        One parameter can be restricted at a time::

            sage: n, k = 250, 70
            sage: codes.decoders.GRSGuruswamiSudanDecoder.guruswami_sudan_decoding_radius(n_k = (n, k), s=3)
            (109, (3, 5))
            sage: codes.decoders.GRSGuruswamiSudanDecoder.guruswami_sudan_decoding_radius(n_k = (n, k), l=7)
            (111, (4, 7))

        The function can also just compute the decoding radius given the parameters::

            sage: codes.decoders.GRSGuruswamiSudanDecoder.guruswami_sudan_decoding_radius(n_k = (n, k), s=2, l=6)
            (92, (2, 6))
        """
        n,k = n_k_params(C, n_k)
        def get_tau(s,l):
            "Return the decoding radius given this s and l"
            if s<=0 or l<=0:
                return -1
            return gilt(n - n/2*(s+1)/(l+1) - (k-1)/2*l/s)
        if l is None and s is None:
            tau = gilt(johnson_radius(n, n - k + 1))
            return (tau, GRSGuruswamiSudanDecoder.parameters_given_tau(tau, n_k = (n, k)))
        if l is not None and s is not None:
            return (get_tau(s,l), (s,l))

        # Either s or l is set, but not both. First a shared local function
        def find_integral_max(real_max, f):
            """Given a real (local) maximum of a function `f`, return that of
            the integers around `real_max` which gives the (local) integral
            maximum, and the value of at that point."""
            if real_max in ZZ:
                int_max = Integer(real_max)
                return (int_max, f(int_max))
            else:
                x_f = floor(real_max)
                x_c = x_f + 1
                f_f, f_c = f(x_f), f(x_c)
                return (x_f, f_f) if f_f >= f_c else (x_c, f_c)

        if s is not None:
            # maximising tau under condition
            # n*(s+1 choose 2) < (ell+1)*s*(n-tau) - (ell+1 choose 2)*(k-1)
            # knowing n and s, we can just minimise
            # ( n*(s+1 choose 2) + (ell+1 choose 2)*(k-1) )/(ell+1)
            # Differentiating and setting to zero yields ell best choice:
            lmax = sqrt(n*s*(s+1.)/(k-1.)) - 1.
            #the best integral value will be
            (l,tau) = find_integral_max(lmax, lambda l: get_tau(s,l))
            #Note that we have not proven that this ell is minimal in integral
            #sense! It just seems that this most often happens
            return (tau,(s,l))
        if l is not None:
            # Acquired similarly to when restricting s
            smax = sqrt((k-1.)/n*l*(l+1.))
            (s,tau) = find_integral_max(smax, lambda s: get_tau(s,l))
            return (tau, (s,l))