By David R. Finston, Patrick J. Morandi

ISBN-10: 3319044974

ISBN-13: 9783319044972

ISBN-10: 3319044982

ISBN-13: 9783319044989

This textual content seeks to generate curiosity in summary algebra by way of introducing every one new constitution and subject through a real-world software. The down-to-earth presentation is offered to a readership without earlier wisdom of summary algebra. scholars are ended in algebraic recommendations and questions in a normal method via their daily studies.

Applications include:

- Identification numbers and modular arithmetic
- (linear) error-correcting codes, together with cyclic codes
- ruler and compass constructions
- cryptography
- symmetry of styles within the actual aircraft

*Abstract Algebra: constitution and Application* is appropriate as a textual content for a primary path on summary algebra whose major function is to generate curiosity within the topic or as a supplementary textual content for extra complicated classes. the cloth paves tips to next classes that additional advance the speculation of summary algebra and should attract scholars of arithmetic, arithmetic schooling, machine technological know-how, and engineering drawn to purposes of algebraic concepts.

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**Extra resources for Abstract Algebra: Structure and Application**

**Example text**

Let C D f00000; 10000; 011000; 11100g. If w D 10001, then w is distance 1 from 10000 and distance more than 1 from the other two codewords. Thus, we would decode w as 10000. However, if u D 11000, then u is distance 1 from both 10000 and from 111000. Thus, either is an appropriate choice to decode u. We now define what it means for a code to be an error correcting code. 4. Let C be a code and let t be a positive integer. Then C is a t-error correcting code if whenever a word w diﬀers from the nearest codeword v by a distance of at most t, then v is the unique closest codeword to w.

4). If one were to write out all 16 codewords in C , one would find the distance of C to be exactly 3. Linear codes like C are identified by their length, dimension, and minimum distance. 7; 4; 3/-code, because its length is 7, its dimension is 4, and its minimum distance is equal to 3. 8 that C corrects 1 error. The code C has a particularly elegant decoding algorithm, which we now describe. Let fe1 ; : : : ; e7 g be the standard basis for Z72 . We point out a fact of matrix multiplication: HeiT is equal to the i th column of H .

Those that contain a leading 1) form a basis for the row space of A. Consequently, the dimension of the row space is the number of nonzero rows in EA . Thus, an alternative definition of the rank of a matrix is the number of leading 1’s in the row reduced echelon form obtained from the matrix. Again these assertions hold for matrices with entries in Z2 . 2 Gaussian Elimination 29 The fact that the homogeneous linear systems AX D 0 and EA X D 0 have the same solutions can be interpreted as the statement that the columns of A and the columns of EA have the identical dependence relations (but their column spaces may be different).

### Abstract Algebra: Structure and Application by David R. Finston, Patrick J. Morandi

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