Galois Field Sequence Combination for Longer Repetition Periods
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Solution Overview
Problem
Existing methods for combining data streams in signal processing, error correction codes, and encryption systems face challenges in increasing the repetition period of sequences efficiently, leading to increased implementation complexity and error proneness.
Innovation Solution
A method involving the reception and mapping of number sequences in a Galois field, followed by Galois field multiplication in a Galois extension field to generate an output sequence, ensuring invertible reconstruction and increased repetition period, using Affine transformations and selective digit multiplication.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Duration of action of stationary object
If conventional sequence combination techniques are used to increase repetition period, then the repetition period increases, but implementation complexity significantly increases
Solution Approach 1:
The patent changes the parameter space by mapping sequences from a base Galois field GF(p^k) to an extended Galois field GF(p^(2k)), thereby increasing the repetition period through field extension rather than through complex combination logic. This parameter change in the mathematical domain achieves the goal with minimal implementation complexity.
Solution Approach 2:
The patent replaces complex mechanical or logical combination mechanisms with algebraic operations in Galois fields. By substituting the combination process with field multiplication and affine transformations, the system achieves sequence combination with reduced complexity while maintaining invertibility.
2Duration of action of stationary object
If sequence length is increased to increase repetition period, then repetition period increases, but processing time increases
Solution Approach 1:
Instead of increasing sequence length linearly, the patent changes the field parameter from p^k to p^(2k), achieving a squared increase in repetition period without proportionally increasing processing time. The algebraic structure of Galois fields allows efficient computation even in extended fields.
Solution Approach 2:
The patent moves the problem from one dimension (sequence length) to another dimension (field extension degree). By operating in an extended Galois field, the system achieves longer repetition periods through a different dimensional approach that is more computationally efficient.
3Duration of action of stationary object
If multiple orthogonal sequences are combined to increase repetition period, then repetition period increases, but resource consumption increases
Solution Approach 1:
The patent creates a universal combination mechanism that works for any two sequences in a Galois field through affine transformations and field multiplication. This single universal approach replaces the need for multiple specialized combination techniques, reducing resource consumption while achieving the same goal.
4Ease of manufacture
If fixed generating algorithms are used for each sequence period, then implementation is simpler, but adaptability decreases
Solution Approach 1:
The patent introduces dynamic elements through time-varying affine transformations while maintaining the overall structure of the algorithm. The transformation parameters can change with time or sequence index, providing adaptability without sacrificing the systematic nature of the implementation.
Data Source
AI summary
A method is provided for combining two or more input sequences in a communications system to increase a repetition period of the input sequences in a resource-efficient manner. The method includes a receiving step, a mapping step, and a generating step. The receiving step involves receiving a first number sequence and a second number sequence, each expressed in a Galois field GF[pk]. The mapping step involves mapping the first and second number sequences to a Galois extension field GF[pk+1]. The generating step involves generating an output sequence by combining the first number sequence with the second number sequence utilizing a Galois field multiplication operation in the Galois extension field GF[pk+1]. p is a prime number. k is an integer. pk+1 defines a finite field size of the Galois extension field GF[pk+1].