Method and apparatus for updating cyclic redundancy check information for data storage

An apparatus for determining whether received data that contains errors has been properly corrected, comprising:

• means for receiving a block of data comprising a data portion, cyclic redundancy check symbols generated using a cyclic redundancy check generator polynomial having a plurality of roots and Reed-Solomon check symbols generated using a Reed-Solomon generator polynomial having a plurality of roots, and including at least one error;
• means for generating at least one cyclic redundancy check syndrome for the received block of data using the cyclic redundancy check generator polynomial;
• means for attempting to correct the at least one error in the block of data;
• means for updating the cyclic redundancy check syndrome to correspond to the corrected block of data;
• and
• means for determining whether the error has been corrected using the updated cyclic redundancy check syndrome;
• wherein the roots of the cyclic redundancy check generator polynomial meet the following conditions:
• (a) the exponents of the roots are distinct values or all values modulo each prime factor of the cycle length of the underlying finite field,
• (b) the roots do not overlap the roots of the Reed-Solomon generator polynomial and any power of two of any root of the Reed-Solomon generator polynomial,
• (c) no root is a power of two of any other root, and
• (d) at least one of the roots is a primitive element of the finite field.

A system for transmitting and receiving data, comprising:

• means for transmitting a block of data comprising a data portion, cyclic redundancy check symbols and Reed-Solomon check symbols, comprising:
• means for generating the cyclic redundancy check symbols using a cyclic redundancy check generator polynomial having a plurality of roots, and
• means for generating the Reed-Solomon check symbols using a Reed-Solomon generator polynomial having a plurality of roots;
• and
• means for determining whether received data that contains errors has been properly corrected, comprising:
• means for receiving the block of data, the block of data including at least one error;
• means for generating at least one cyclic redundancy check syndrome for the received block of data using the cyclic redundancy check generator polynomial;
• means for attempting to correct the at least one error in the block of data;
• means for updating the cyclic redundancy check syndrome to correspond to the corrected block of data;
• and
• means for determining whether the error has been corrected using the updated cyclic redundancy check syndrome;
• wherein the roots of the cyclic redundancy check generator polynomial meet the following conditions:
• (a) the exponents of the roots are distinct values or all values modulo each prime factor of the cycle length of the underlying finite field,
• (b) the roots do not overlap the roots of the Reed-Solomon generator polynomial and any power of two of any root of the Reed-Solomon generator polynomial,
• (c) no root is a power of two of any other root, and
• (d) at least one of the roots is a primitive element of the finite field.

In a system for transmItting and receiving data, an apparatus for transmitting a block of data comprising a data portion, cyclic redundancy check symbols and Reed-Solomon check symbols, comprising:

• means for generating the cyclic redundancy check symbols using a cyclic redundancy check generator polynomial having a plurality of roots;
• and
• means for generating the Reed-Solomon check symbols using a Reed-Solomon generator polynomial having a plurality of roots;
• wherein the roots of the cyclic redundancy check generator polynomial meet the following conditions:
• (a) the exponents of the roots are distinct values or all values modulo each prime factor of the cycle length of the underlying finite field,
• (b) the roots do not overlap the roots of the Reed-Solomon generator polynomial and any power of two of any root of the Reed-Solomon generator polynomial,
• (c) no root is a power of two of any other root, and
• (d) at least one of the roots is a primitive element of the finite field.

An apparatus for determining whether received data that contains errors have been properly corrected, comprising:

• a receiver operable to receive a block of data comprising a data portion, cyclic redundancy check symbols generated using a cyclic redundancy check generator polynomial having a plurality of roots and Reed-Solomon check symbols generated using a Reed-Solomon generator polynomial having a plurality of roots, and including at least one error;
• a cyclic redundancy check syndrome generator operable to generate at least one cyclic redundancy check syndrome for the received block of data using the cyclic redundancy check generator polynomial;
• a device operable to attempt to correct the at least one error in the block of data;
• a device operable to update the cyclic redundancy check syndrome to correspond to the corrected block of data;
• and
• a device operable to determine whether the error has been corrected using the updated cyclic redundancy check syndrome;
• wherein the roots of the cyclic redundancy check generator polynomial meet the following conditions:
• (a) the exponents of the roots are distinct values or all values modulo each prime factor of the cycle length of the underlying finite field,
• (b) the roots do not overlap the roots of the Reed-Solomon generator polynomial and any power of two of any root of the Reed-Solomon generator polynomial,
• (c) no root is a power of two of any other root, and
• (d) at least one of the roots is a primitive element of the finite field.

A system for transmitting and receiving data, comprising:

• a device operable to transmit a block of data comprising a data portion, cyclic redundancy check symbols and Reed-Solomon check symbols, comprising:
• a cyclic redundancy code generator operable to generate the cyclic redundancy check symbols using a cyclic redundancy check generator polynomial having a plurality of roots, and
• a Reed-Solomon code generator operable to generate the Reed-Solomon check symbols using a Reed-Solomon generator polynomial having a plurality of roots;
• and
• a device operable to determining whether received data that contains errors has been properly corrected, comprising:
• a receiver operable to receive the block of data, the block of data including at least one error;
• a cyclic redundancy check syndrome generator operable to generate at least one cyclic redundancy check syndrome for the received block of data using the cyclic redundancy check generator polynomial;
• a device operable to attempt to correct the at least one error in the block of data;
• a device operable to update the cyclic redundancy check syndrome to correspond to the corrected block of data;
• and
• a device operable to determine whether the error has been corrected using the updated cyclic redundancy check syndrome;
• wherein the roots of the cyclic redundancy check generator polynomial meet the following conditions:
• (a) the exponents of the roots are distinct values or all values modulo each prime factor of the cycle length of the underlying finite field,
• (b) the roots do not overlap the roots of the Reed-Solomon generator polynomial and any power of two of any root of the Reed-Solomon generator polynomial,
• (c) no root is a power of two of any other root, and
• (d) at least one of the roots is a primitive element of the finite field.

In a system for transmitting and receiving data, an apparatus for transmitting a block of data comprising a data portion, cyclic redundancy check symbols and Reed-Solomon check symbols, comprising:

• a cyclic redundancy code generator operable to generate the cyclic redundancy check symbols using a cyclic redundancy check generator polynomial having a plurality of roots;
• and
• a Reed-Solomon code generator operable to generate the Reed-Solomon check symbols using a Reed-Solomon generator polynomial having a plurality of roots;
• wherein the roots of the cyclic redundancy check generator polynomial meet the following conditions:
• (a) the exponents of the roots are distinct values or all values modulo each prime factor of the cycle length of the underlying finite field,
• (b) the roots do not overlap the roots of the Reed-Solomon generator polynomial and any power of two of any root of the Reed-Solomon generator polynomial,
• (c) no root is a power of two of any other root, and
• (d) at least one of the roots is a primitive element of the finite field.

A medium, executable in a computer system, for determining whether received data that contained errors has been properly corrected, comprising:

• program instructions for receiving a block of data comprising a data portion, cyclic redundancy check symbols generated using a cyclic redundancy check generator polynomial having a plurality of roots and Reed-Solomon check symbols generated using a Reed-Solomon generator polynomial having a plurality of roots, and including at least one error;
• program instructions for generating at least one cyclic redundancy check syndrome for the received block of data using the cyclic redundancy check generator polynomial;
• attempting to correct the at least one error in the block of data;
• program instructions for updating the cyclic redundancy check syndrome to correspond to the corrected block of data;
• and
• program instructions for determining whether the error has been corrected using the updated cyclic redundancy check syndrome;
• wherein the roots of the cyclic redundancy check generator polynomial meet the following conditions:
• (a) the exponents of the roots are distinct values or all values modulo each prime factor of the cycle length of the underlying finite field,
• (b) the roots do not overlap the roots of the Reed-Solomon generator polynomial and any power of two of any root of the Reed-Solomon generator polynomial,
• (c) no root is a power of two of any other root, and
• (d) at least one of the roots is a primitive element of the finite field.