Improved Modular Division Implementation with the Akushsky Core Function

The residue number system (RNS) is widely used in different areas due to the efficiency of modular addition and multiplication operations. However, non-modular operations, such as sign and division operations, are computationally complex. A fractional representation based on the Chinese remainder theorem is widely used. In some cases, this method gives an incorrect result associated with round-off calculation errors. In this paper, we optimize the division operation in RNS using the Akushsky core function without critical cores. We show that the proposed method reduces the size of the operands by half and does not require additional restrictions on the divisor as in the division algorithm in RNS based on the approximate method.

Encoder Lipschitz Integers: The Lipschitz integers that have the ”division with small remainder” property

The residue class set of a Lipschitz integer is constructed by modulo function with primitive Lipschitz integer whose norm is a prime integer, i.e. prime Lipschitz integer. In this study, we consider primitive Lipschitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Lipschitz integer is less than the norm of the primitive Lipschitz integer used to construct the residue class set of the Lipschitz integer, then, the Euclid division algorithm works for this primitive Lipschitz integer. The Euclid division algorithm always works for prime Lipschitz integers. In other words, the prime Lipschitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Lipschitz residue class set that lies on primitive Lipschitz integers whose norm is not a prime integer. In this study, we solve this problem by defining Lipschitz integers that have the ”division with small remainder” property, namely, encoder Lipschitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Also, we investigate the performances of Lipschitz signal constellations (the left residue class set) obtained by modulo function with Lipschitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by agency of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Encoder Hurwitz Integers: The Hurwitz integers that have the ”division with small remainder” property

The residue class set of a Hurwitz integer is constructed by modulo function with primitive Hurwitz integer whose norm is a prime integer, i.e. prime Hurwitz integer. In this study, we consider primitive Hurwitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Hurwitz integer is less than the norm of the primitive Hurwitz integer used to construct the residue class set of the Hurwitz integer, then, the Euclid division algorithm works for this primitive Hurwitz integer. The Euclid division algorithm always works for prime Hurwitz integers. In other words, the prime Hurwitz integers and halves-integer primitive Hurwitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Hurwitz residue class set that lies on primitive Hurwitz integers that their norms are not a prime integer and their components are in integers set. In this study, we solve this problem by defining Hurwitz integers that have the ”division with small remainder” property, namely, encoder Hurwitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Especially, Euclidean metric. Also, we investigate the performances of Hurwitz signal constellations (the left residue class set) obtained by modulo function with Hurwitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by means of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Encoder Lipschitz Integers: The Lipschitz integers that have the ”division with small remainder” property

The residue class set of a Lipschitz integer is constructed by modulo function with primitive Lipschitz integer whose norm is a prime integer, i.e. prime Lipschitz integer. In this study, we consider primitive Lipschitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Lipschitz integer is less than the norm of the primitive Lipschitz integer used to construct the residue class set of the Lipschitz integer, then, the Euclid division algorithm works for this primitive Lipschitz integer. The Euclid division algorithm always works for prime Lipschitz integers. In other words, the prime Lipschitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Lipschitz residue class set that lies on primitive Lipschitz integers whose norm is not a prime integer. In this study, we solve this problem by defining Lipschitz integers that have the ”division with small remainder” property, namely, encoder Lipschitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Also, we investigate the performances of Lipschitz signal constellations (the left residue class set) obtained by modulo function with Lipschitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by agency of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Encoder Hurwitz Integers: The Hurwitz integers that have the ”division with small remainder” property

The residue class set of a Hurwitz integer is constructed by modulo function with primitive Hurwitz integer whose norm is a prime integer, i.e. prime Hurwitz integer. In this study, we consider primitive Hurwitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Hurwitz integer is less than the norm of the primitive Hurwitz integer used to construct the residue class set of the Hurwitz integer, then, the Euclid division algorithm works for this primitive Hurwitz integer. The Euclid division algorithm always works for prime Hurwitz integers. In other words, the prime Hurwitz integers and halves-integer primitive Hurwitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Hurwitz residue class set that lies on primitive Hurwitz integers that their norms are not a prime integer and their components are in integers set. In this study, we solve this problem by defining Hurwitz integers that have the ”division with small remainder” property, namely, encoder Hurwitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Especially, Euclidean metric. Also, we investigate the performances of Hurwitz signal constellations (the left residue class set) obtained by modulo function with Hurwitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by means of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Encoder Hurwitz Integers: The Hurwitz integers that have the ”division with small division” property

The residue class set of a Hurwitz integer is constructed by modulo function with primitive Hurwitz integer whose norm is a prime integer, i.e. prime Hurwitz integer. In this study, we consider primitive Hurwitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Hurwitz integer is less than the norm of the primitive Hurwitz integer used to construct the residue class set of the Hurwitz integer, then, the Euclid division algorithm works for this primitive Hurwitz integer. The Euclid division algorithm always works for prime Hurwitz integers. In other words, the prime Hurwitz integers and halves-integer primitive Hurwitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Hurwitz residue class set that lies on primitive Hurwitz integers that their norms are not a prime integer and their components are in integers set. In this study, we solve this problem by defining Hurwitz integers that have the ”division with small remainder” property, namely, encoder Hurwitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Especially, Euclidean metric. Also, we investigate the performances of Hurwitz signal constellations (the left residue class set) obtained by modulo function with Hurwitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by means of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Encoder Hurwitz Integers: The Hurwitz integers that have the ”division with small division” property

The residue class set of a Hurwitz integer is constructed by modulo function with primitive Hurwitz integer whose norm is a prime integer, i.e. prime Hurwitz integer. In this study, we consider primitive Hurwitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Hurwitz integer is less than the norm of the primitive Hurwitz integer used to construct the residue class set of the Hurwitz integer, then, the Euclid division algorithm works for this primitive Hurwitz integer. The Euclid division algorithm always works for prime Hurwitz integers. In other words, the prime Hurwitz integers and halves-integer primitive Hurwitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Hurwitz residue class set that lies on primitive Hurwitz integers that their norms are not a prime integer and their components are in integers set. In this study, we solve this problem by defining Hurwitz integers that have the ”division with small remainder” property, namely, encoder Hurwitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Especially, Euclidean metric. Also, we investigate the performances of Hurwitz signal constellations (the left residue class set) obtained by modulo function with Hurwitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by means of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Encoder Lipschitz Integers: The Lipschitz integers that have the ”division with small division” property

The residue class set of a Lipschitz integer is constructed by modulo function with primitive Lipschitz integer whose norm is a prime integer, i.e. prime Lipschitz integer. In this study, we consider primitive Lipschitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Lipschitz integer is less than the norm of the primitive Lipschitz integer used to construct the residue class set of the Lipschitz integer, then, the Euclid division algorithm works for this primitive Lipschitz integer. The Euclid division algorithm always works for prime Lipschitz integers. In other words, the prime Lipschitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Lipschitz residue class set that lies on primitive Lipschitz integers whose norm is not a prime integer. In this study, we solve this problem by defining Lipschitz integers that have the ”division with small remainder” property, namely, encoder Lipschitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Also, we investigate the performances of Lipschitz signal constellations (the left residue class set) obtained by modulo function with Lipschitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by agency of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).