Abstract: A decoding method decodes N received branchwords produced by a convolutional encoder using a tail-biting convolutional code. N received branchwords are in memory. Viterbi updates are performed on a sequence of branchwords. A first encoder determines a state at the end of the third block most likely to have generated the final branchword in the sequence from the best path metric. A Viterbi traceback procedure is performed from that first encoder state at the end of the third block to determine a second encoder state at the start of the third block of branchwords. A Viterbi traceback procedure is performed from that second encoder state at the start of the third block to determine a third encoder state at the start of the second block of branchwords. A derived tail-biting path is output, if the second and third encoder states are identical.

Abstract: The decoding method comprises storing, successively performing, determining, first performing, second performing, and outputting. The storing stores the N received branchwords in memory. The successively performing performs Viterbi updates on a sequence of branchwords. The determining determines a first encoder state at the end of the third block most likely to have generated the final branchword in the sequence from the best path metric. The first performing performs a Viterbi traceback procedure from the first encoder state at the end of the third block to determine a second encoder state at the start of the third block of branchwords. The second performing performs a Viterbi traceback procedure from that second encoder state at the start of the third block to determine a third encoder state at the start of the second block of branchwords. The outputting outputs a derived tail-biting path, if the second and third encoder states are identical.

Abstract: Provided herein are methods of increasing the enzymatic rate of hydrolysis of starch substrates. In certain embodiments the method comprises contacting a starch substrate with one or more amylase family enzymes in the presence of greater than 0.001 mM manganese ions. In certain embodiments, the method comprises contacting a starch substrate with one or more glucoamylase and/or ?-amylase enzymes in the presence of greater than 0.001 mM calcium ions (Ca++). In certain embodiments, the method comprises contacting the starch substrate with a ?-amylase in the presence of greater than 0.001 mM manganese ion, calcium ion, magnesium ion (Mg++), strontium ion (Sr++), barium ion (Ba++) or any combination of said metal ions. In certain embodiments the method comprises contacting the starch substrate with a glucoamylase in the presence of greater than 0.001 mM manganese ion, calcium ion, lithium ion (Li+), potassium ion (K+), or any combination of said metal ions.