Fast efficient decoder for low distance RS and BCH codes

An error correction method that comprises:

• obtaining syndrome values corresponding to a product of a receive message vector and a parity check matrix, the syndrome values including at least a first ten-bit syndrome value S1, a second ten-bit element syndrome value S3, and a third ten-bit syndrome value S5;
• converting the syndrome values into a set of zero or more polynomial roots corresponding to error locations in the receive message vector, said converting including:
• deriving a number of errors from the syndrome values;
• and
• when the number of errors is two, using an array of logic gates to obtain two polynomial roots corresponding to a product of a syndrome value vector and a quadratic solution matrix, the quadratic solution matrix corresponding to a determination of a quadratic equation's trailing coefficient value s, a determination of the quadratic equation's roots, and a reversal of a variable substitution;
• and
• determining a bit index for each polynomial root in the set, the bit indexes indicating the error locations to be corrected in the receive message vector.

An error correction method that comprises:

• obtaining syndrome values corresponding to a product of a receive message vector and a parity check matrix, the syndrome values including at least a first ten-bit syndrome value S1, a second ten-bit element syndrome value S3, and a third ten-bit syndrome value S5;
• converting the syndrome values into a set of zero or more polynomial roots corresponding to error locations in the receive message vector, said converting including:
• deriving a number of errors from the syndrome values;
• and
• when the number of errors is three, using a first lookup table with supporting logic gates to obtain three polynomial roots when
• (
• S
• 5
• +
• S
• 1
• 5
• )
• =
• 0
• ,
• the three polynomial roots corresponding to roots of a cubic equation x3+d with d representable as
• S
• 3
• +
• S
• 1
• 3
• ,
• the first lookup table having a depth of no more than 341;
• and
• determining a bit index for each polynomial root in the set, the bit indexes indicating the error locations to be corrected in the receive message vector.